A Lean Companion to Conceptual Mathematics

Session 24: Uniqueness of products and definition of sum🔗

2. The uniqueness theorem for products🔗

Theorem (uniqueness of products) (p. 263)

Suppose that both of B_1 \xleftarrow{p_1} P \xrightarrow{p_2} B_2 \quad\text{and}\quad {B_1 \xleftarrow{q_1} Q \xrightarrow{q_2} B_2} are product projection pairs (i.e. the ps as well as the qs satisfy the universal mapping property). Then there is exactly one map {P \xrightarrow{f} Q} for which q_1 f = p_1 \quad\text{and}\quad q_2 f = p_2 This map f is in fact an isomorphism.

See the previous formalisations provided in Exercise 12 of Article IV and on pp. 221 & 239.

3. Sum of two objects in a category🔗

Definition: Sum of two objects (p. 265–266)

A sum of two objects B_1, B_2 is an object S and a pair of maps B_1 \xrightarrow{j_1} S \xleftarrow{j_2} B_2 having the following universal mapping property: For any two maps B_1 \xrightarrow{f_1} X \xleftarrow{f_2} B_2 among all the maps {X \xleftarrow{f} S} there is exactly one that satisfies both f_1 = f j_1 \quad\text{and}\quad f_2 = f j_2 ...

See the earlier formalisation of sum on p. 222.

Exercise 1 (p. 266)

Formulate and prove the theorem of uniqueness of sums.

Solution: Exercise 1

Let S and T be two sums of B_1 and B_2, with injection maps {B_1 \xrightarrow{j_1} S \xleftarrow{j_2} B_2} and {B_1 \xrightarrow{k_1} T \xleftarrow{k_2} B_2}.

We first provide a proof of the theorem using the mathlib API for (binary) coproducts. For symmetry with our earlier proof of the uniqueness of products, we include the additional result that the unique map between any two sums of the same two objects in a category is an isomorphism.

From the formalisation in mathlib, we obtain the following mappings between the notation for this exercise given above and the data used in our proof:

S = coconeS.pt; j₁ = coconeS.ι.app ⟨WalkingPair.left⟩; j₂ = coconeS.ι.app ⟨WalkingPair.right⟩

T = coconeT.pt; k₁ = coconeT.ι.app ⟨WalkingPair.left⟩; k₂ = coconeT.ι.app ⟨WalkingPair.right⟩

theorem uniqueness_of_sums {𝒞 : Type u} [Category.{v, u} 𝒞] (B₁ B₂ : 𝒞) (coconeS coconeT : BinaryCofan B₁ B₂) (sumS : IsColimit coconeS) (sumT : IsColimit coconeT) : ∃! f : coconeS.pt ⟶ coconeT.pt, (coconeT.ι.app ⟨WalkingPair.left⟩ = f ⊚ coconeS.ι.app ⟨WalkingPair.left⟩ ∧ coconeT.ι.app ⟨WalkingPair.right⟩ = f ⊚ coconeS.ι.app ⟨WalkingPair.right⟩) ∧ IsIso f := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∃! f, (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) (sumS.desc coconeT) ∧ ∀ (y : coconeS.pt ⟶ coconeT.pt), (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) y → y = sumS.desc coconeT -- f 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.left } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.left }𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.right } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.right }𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ IsIso (sumS.desc coconeT)𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∀ (y : coconeS.pt ⟶ coconeT.pt), (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) y → y = sumS.desc coconeT 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.left } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.left } -- Show that k₁ = f ⊚ j₁ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.right } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.right } -- Show that k₂ = f ⊚ j₂ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ IsIso (sumS.desc coconeT) -- Show that f is an isomorphism 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∃ inv, inv ⊚ sumS.desc coconeT = 𝟙 coconeS.pt ∧ sumS.desc coconeT ⊚ inv = 𝟙 coconeT.pt 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumT.desc coconeS ⊚ sumS.desc coconeT = 𝟙 coconeS.pt ∧ sumS.desc coconeT ⊚ sumT.desc coconeS = 𝟙 coconeT.pt -- f⁻¹ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumT.desc coconeS ⊚ sumS.desc coconeT = 𝟙 coconeS.pt𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumS.desc coconeT ⊚ sumT.desc coconeS = 𝟙 coconeT.pt 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumT.desc coconeS ⊚ sumS.desc coconeT = 𝟙 coconeS.pt -- f⁻¹ ⊚ f = 𝟙 S 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumT.desc coconeS ⊚ sumS.desc coconeT = sumS.desc coconeS 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∀ (j : Discrete WalkingPair), (sumT.desc coconeS ⊚ sumS.desc coconeT) ⊚ coconeS.ι.app j = coconeS.ι.app j 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTj:Discrete WalkingPair⊢ (sumT.desc coconeS ⊚ sumS.desc coconeT) ⊚ coconeS.ι.app j = coconeS.ι.app j erw [← Category.assoc, sumS.fac, sumT.facAll goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumS.desc coconeT ⊚ sumT.desc coconeS = 𝟙 coconeT.pt -- f ⊚ f⁻¹ = 𝟙 T 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ sumS.desc coconeT ⊚ sumT.desc coconeS = sumT.desc coconeT 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∀ (j : Discrete WalkingPair), (sumS.desc coconeT ⊚ sumT.desc coconeS) ⊚ coconeT.ι.app j = coconeT.ι.app j 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTj:Discrete WalkingPair⊢ (sumS.desc coconeT ⊚ sumT.desc coconeS) ⊚ coconeT.ι.app j = coconeT.ι.app j erw [← Category.assoc, sumT.fac, sumS.facAll goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∀ (y : coconeS.pt ⟶ coconeT.pt), (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) y → y = sumS.desc coconeT -- Show that f is unique intro f 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f = sumS.desc coconeT 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ ∀ (j : Discrete WalkingPair), f ⊚ coconeS.ι.app j = coconeT.ι.app j 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso fj:Discrete WalkingPair⊢ f ⊚ coconeS.ι.app j = coconeT.ι.app j 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.left } = coconeT.ι.app { as := WalkingPair.left }𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.right } = coconeT.ι.app { as := WalkingPair.right } 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.left } = coconeT.ι.app { as := WalkingPair.left } All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.right } = coconeT.ι.app { as := WalkingPair.right } All goals completed! 🐙

We can use the mathlib lemma IsColimit.nonempty_isColimit_iff_isIso_desc to do much of the heavy lifting here. A revised version of the proof using this lemma is given below.

example {𝒞 : Type u} [Category.{v, u} 𝒞] (B₁ B₂ : 𝒞) (coconeS coconeT : BinaryCofan B₁ B₂) (sumS : IsColimit coconeS) (sumT : IsColimit coconeT) : ∃! f : coconeS.pt ⟶ coconeT.pt, (coconeT.ι.app ⟨WalkingPair.left⟩ = f ⊚ coconeS.ι.app ⟨WalkingPair.left⟩ ∧ coconeT.ι.app ⟨WalkingPair.right⟩ = f ⊚ coconeS.ι.app ⟨WalkingPair.right⟩) ∧ IsIso f := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∃! f, (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) (sumS.desc coconeT) ∧ ∀ (y : coconeS.pt ⟶ coconeT.pt), (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) y → y = sumS.desc coconeT 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.left } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.left }𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.right } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.right }𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ IsIso (sumS.desc coconeT)𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∀ (y : coconeS.pt ⟶ coconeT.pt), (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) y → y = sumS.desc coconeT 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.left } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.left } -- Show that k₁ = f ⊚ j₁ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ coconeT.ι.app { as := WalkingPair.right } = sumS.desc coconeT ⊚ coconeS.ι.app { as := WalkingPair.right } -- Show that k₂ = f ⊚ j₂ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ IsIso (sumS.desc coconeT) -- Show that f is an isomorphism All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeT⊢ ∀ (y : coconeS.pt ⟶ coconeT.pt), (fun f ↦ (coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left } ∧ coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }) ∧ IsIso f) y → y = sumS.desc coconeT -- Show that f is unique intro f 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f = sumS.desc coconeT 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ ∀ (j : Discrete WalkingPair), f ⊚ coconeS.ι.app j = coconeT.ι.app j 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso fj:Discrete WalkingPair⊢ f ⊚ coconeS.ι.app j = coconeT.ι.app j 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.left } = coconeT.ι.app { as := WalkingPair.left }𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.right } = coconeT.ι.app { as := WalkingPair.right } 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.left } = coconeT.ι.app { as := WalkingPair.left } All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞coconeS:BinaryCofan B₁ B₂coconeT:BinaryCofan B₁ B₂sumS:IsColimit coconeSsumT:IsColimit coconeTf:coconeS.pt ⟶ coconeT.pth_left:coconeT.ι.app { as := WalkingPair.left } = f ⊚ coconeS.ι.app { as := WalkingPair.left }h_right:coconeT.ι.app { as := WalkingPair.right } = f ⊚ coconeS.ι.app { as := WalkingPair.right }right✝:IsIso f⊢ f ⊚ coconeS.ι.app { as := WalkingPair.right } = coconeT.ι.app { as := WalkingPair.right } All goals completed! 🐙

We next provide a proof using only the book definition of sum.

theorem uniqueness_of_sums' {𝒞 : Type u} [Category.{v, u} 𝒞] (B₁ B₂ S T : 𝒞) (j₁ : B₁ ⟶ S) (j₂ : B₂ ⟶ S) (hS : ∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), (∃! f : S ⟶ X, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂)) (k₁ : B₁ ⟶ T) (k₂ : B₂ ⟶ T) (hT : ∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), (∃! f : T ⟶ X, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂)) : Nonempty (S ≅ T) := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂⊢ Nonempty (S ≅ T) 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = f⊢ Nonempty (S ≅ T) 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ Nonempty (S ≅ T) have hgf : g ⊚ f = 𝟙 S := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂⊢ Nonempty (S ≅ T) 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = (g ⊚ f) ⊚ j₁ ∧ j₂ = (g ⊚ f) ⊚ j₂𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = 𝟙 S ⊚ j₁ ∧ j₂ = 𝟙 S ⊚ j₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = (g ⊚ f) ⊚ j₁ ∧ j₂ = (g ⊚ f) ⊚ j₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = (g ⊚ f) ⊚ j₁𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₂ = (g ⊚ f) ⊚ j₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = (g ⊚ f) ⊚ j₁ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ (g ⊚ f) ⊚ j₁ = j₁ calc (g ⊚ f) ⊚ j₁ _ = g ⊚ (f ⊚ j₁) := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ (g ⊚ f) ⊚ j₁ = g ⊚ f ⊚ j₁ All goals completed! 🐙 _ = g ⊚ k₁ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ g ⊚ f ⊚ j₁ = g ⊚ k₁ All goals completed! 🐙 _ = j₁ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ g ⊚ k₁ = j₁ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₂ = (g ⊚ f) ⊚ j₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ (g ⊚ f) ⊚ j₂ = j₂ calc (g ⊚ f) ⊚ j₂ _ = g ⊚ (f ⊚ j₂) := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ (g ⊚ f) ⊚ j₂ = g ⊚ f ⊚ j₂ All goals completed! 🐙 _ = g ⊚ k₂ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ g ⊚ f ⊚ j₂ = g ⊚ k₂ All goals completed! 🐙 _ = j₂ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ g ⊚ k₂ = j₂ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = 𝟙 S ⊚ j₁ ∧ j₂ = 𝟙 S ⊚ j₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = 𝟙 S ⊚ j₁𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₂ = 𝟙 S ⊚ j₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₁ = 𝟙 S ⊚ j₁𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = g⊢ j₂ = 𝟙 S ⊚ j₂ All goals completed! 🐙 have hfg : f ⊚ g = 𝟙 T := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂⊢ Nonempty (S ≅ T) 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = (f ⊚ g) ⊚ k₁ ∧ k₂ = (f ⊚ g) ⊚ k₂𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = 𝟙 T ⊚ k₁ ∧ k₂ = 𝟙 T ⊚ k₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = (f ⊚ g) ⊚ k₁ ∧ k₂ = (f ⊚ g) ⊚ k₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = (f ⊚ g) ⊚ k₁𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₂ = (f ⊚ g) ⊚ k₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = (f ⊚ g) ⊚ k₁ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ (f ⊚ g) ⊚ k₁ = k₁ calc (f ⊚ g) ⊚ k₁ _ = f ⊚ (g ⊚ k₁) := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ (f ⊚ g) ⊚ k₁ = f ⊚ g ⊚ k₁ All goals completed! 🐙 _ = f ⊚ j₁ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ f ⊚ g ⊚ k₁ = f ⊚ j₁ All goals completed! 🐙 _ = k₁ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ f ⊚ j₁ = k₁ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₂ = (f ⊚ g) ⊚ k₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ (f ⊚ g) ⊚ k₂ = k₂ calc (f ⊚ g) ⊚ k₂ _ = f ⊚ (g ⊚ k₂) := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ (f ⊚ g) ⊚ k₂ = f ⊚ g ⊚ k₂ All goals completed! 🐙 _ = f ⊚ j₂ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ f ⊚ g ⊚ k₂ = f ⊚ j₂ All goals completed! 🐙 _ = k₂ := 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ f ⊚ j₂ = k₂ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = 𝟙 T ⊚ k₁ ∧ k₂ = 𝟙 T ⊚ k₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = 𝟙 T ⊚ k₁𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₂ = 𝟙 T ⊚ k₂ 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₁ = 𝟙 T ⊚ k₁𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 S⊢ k₂ = 𝟙 T ⊚ k₂ All goals completed! 🐙 𝒞:Type uinst✝:Category.{v, u} 𝒞B₁:𝒞B₂:𝒞S:𝒞T:𝒞j₁:B₁ ⟶ Sj₂:B₂ ⟶ ShS:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂k₁:B₁ ⟶ Tk₂:B₂ ⟶ ThT:∀ (X : 𝒞) (f₁ : B₁ ⟶ X) (f₂ : B₂ ⟶ X), ∃! f, f₁ = f ⊚ k₁ ∧ f₂ = f ⊚ k₂f:S ⟶ Thf_comm:k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂right✝¹:∀ (y : S ⟶ T), (fun f ↦ k₁ = f ⊚ j₁ ∧ k₂ = f ⊚ j₂) y → y = fg:T ⟶ Shg_comm:j₁ = g ⊚ k₁ ∧ j₂ = g ⊚ k₂right✝:∀ (y : T ⟶ S), (fun f ↦ j₁ = f ⊚ k₁ ∧ j₂ = f ⊚ k₂) y → y = ghgf:g ⊚ f = 𝟙 Shfg:f ⊚ g = 𝟙 T⊢ S ≅ T All goals completed! 🐙
Exercise 2 (p. 267)

Prove the following formulas:

(a) {D + D = 2 \times D}

(b) {D × D = D}

(c) {A × D = D + D}

Solution: Exercise 2

For part (a), we first formalise the graph 2.

def IG2 : IrreflexiveGraph := { carrierA := Fin 2 carrierD := Fin 2 toSrc := fun x ↦ x toTgt := fun x ↦ x }

We then show that {D + D = 2 \times D} (up to isomorphism).

open IrreflexiveGraph in noncomputable example {sumDD prod2D : IrreflexiveGraph} (j₁ : D ⟶ sumDD) (j₂ : D ⟶ sumDD) (h_sumDD : ∀ (X : IrreflexiveGraph) (f₁ : D ⟶ X) (f₂ : D ⟶ X), (∃! f : sumDD ⟶ X, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂)) (p₁ : prod2D ⟶ IG2) (p₂ : prod2D ⟶ D) (h_prod2D : ∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), (∃! g : Y ⟶ prod2D, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g)) : sumDD ≅ prod2D := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ sumDD ≅ prod2D -- Define graph G isomorphic to both sumDD and prod2D sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ sumDD ≅ prod2D -- Define injections to G let jG₁ : D ⟶ G := { val := ⟨Empty.elim, fun _ ↦ 0⟩ property := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 0).1 ∧ (Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 0).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 0).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1 (sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }x:D.carrierA⊢ ((Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt) x = (G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1) x; All goals completed! 🐙) } let jG₂ : D ⟶ G := { val := ⟨Empty.elim, fun _ ↦ 1⟩ property := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 1).1 ∧ (Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 1).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 1).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1 (sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩x:D.carrierA⊢ ((Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt) x = (G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1) x; All goals completed! 🐙) } -- Define projections from G let pG₁ : G ⟶ IG2 := { val := ⟨Empty.elim, fun x ↦ x⟩ property := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toSrc = IG2.toSrc ⊚ (Empty.elim, fun x ↦ x).1 ∧ (Empty.elim, fun x ↦ x).2 ⊚ G.toTgt = IG2.toTgt ⊚ (Empty.elim, fun x ↦ x).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toSrc = IG2.toSrc ⊚ (Empty.elim, fun x ↦ x).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toTgt = IG2.toTgt ⊚ (Empty.elim, fun x ↦ x).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toSrc = IG2.toSrc ⊚ (Empty.elim, fun x ↦ x).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toTgt = IG2.toTgt ⊚ (Empty.elim, fun x ↦ x).1 (sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩x:G.carrierA⊢ ((Empty.elim, fun x ↦ x).2 ⊚ G.toTgt) x = (IG2.toTgt ⊚ (Empty.elim, fun x ↦ x).1) x; All goals completed! 🐙) } let pG₂ : G ⟶ D := { val := ⟨Empty.elim, fun x ↦ ()⟩ property := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toSrc = D.toSrc ⊚ (Empty.elim, fun x ↦ ()).1 ∧ (Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt = D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toSrc = D.toSrc ⊚ (Empty.elim, fun x ↦ ()).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt = D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toSrc = D.toSrc ⊚ (Empty.elim, fun x ↦ ()).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt = D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1 (sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩x:G.carrierA⊢ ((Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt) x = (D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1) x; All goals completed! 🐙) } -- Show that G satisfies universal property for sums have h_sumG : ∀ (X : IrreflexiveGraph) (f₁ : D ⟶ X) (f₂ : D ⟶ X), (∃! f : G ⟶ X, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ sumDD ≅ prod2D sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂ use { val := ⟨Empty.elim, fun | 0 => f₁.val.2 () | 1 => f₂.val.2 ()⟩ property := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 ∧ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 (sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:G.carrierA⊢ ((Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt) x = (X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1) x; All goals completed! 🐙) } sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ ∀ (y : G ⟶ X), (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) y → y = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ f₁ = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₁sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ f₂ = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂ all_goals sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:D.carrierA⊢ (↑f₂).1 x = (↑(⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂)).1 xsumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:D.carrierD⊢ (↑f₂).2 x = (↑(⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂)).2 x sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:D.carrierD⊢ (↑f₂).2 x = (↑(⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂)).2 x All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ ∀ (y : G ⟶ X), (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) y → y = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ f = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierA⊢ (↑f).1 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).1 xsumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierD⊢ (↑f).2 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 x sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierA⊢ (↑f).1 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).1 x All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierD⊢ (↑f).2 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 x sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨0, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨0, ⋯⟩)sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨1, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨1, ⋯⟩) sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨0, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨0, ⋯⟩) sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨0, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑(f ⊚ jG₁)).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨0, ⋯⟩); All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨1, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨1, ⋯⟩) sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩X:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨1, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑(f ⊚ jG₂)).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨1, ⋯⟩); All goals completed! 🐙 -- Hence G is isomorphic to sumDD by uniqueness of sums sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDD⊢ sumDD ≅ prod2D -- Show that G satisfies universal property for products have h_prodG : ∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), (∃! g : Y ⟶ G, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ sumDD ≅ prod2D sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g use { val := ⟨g₂.val.1, g₁.val.2⟩ property := sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toSrc = G.toSrc ⊚ ((↑g₂).1, (↑g₁).2).1 ∧ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt = G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toSrc = G.toSrc ⊚ ((↑g₂).1, (↑g₁).2).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt = G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toSrc = G.toSrc ⊚ ((↑g₂).1, (↑g₁).2).1sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt = G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1 (sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ De:Y.carrierA⊢ (((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt) e = (G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1) e; All goals completed! 🐙) } sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) ⟨((↑g₂).1, (↑g₁).2), ⋯⟩sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ∀ (y : Y ⟶ G), (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) y → y = ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ g₁ = pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ g₂ = pG₂ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ g₁ = pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ De:Y.carrierA⊢ (↑g₁).1 e = (↑(pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩)).1 esumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ De:Y.carrierD⊢ (↑g₁).2 e = (↑(pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩)).2 e sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ De:Y.carrierA⊢ (↑g₁).1 e = (↑(pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩)).1 e All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ De:Y.carrierD⊢ (↑g₁).2 e = (↑(pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩)).2 e All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ g₂ = pG₂ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ De:Y.carrierA⊢ (↑g₂).1 e = (↑(pG₂ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩)).1 e All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ D⊢ ∀ (y : Y ⟶ G), (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) y → y = ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g⊢ g = ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierA⊢ (↑g).1 e = (↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).1 esumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierD⊢ ↑((↑g).2 e) = ↑((↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).2 e) sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierA⊢ (↑g).1 e = (↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).1 e All goals completed! 🐙 sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierD⊢ ↑((↑g).2 e) = ↑((↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).2 e) sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDY:IrreflexiveGraphg₁:Y ⟶ IG2g₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierD⊢ ↑((↑g).2 e) = ↑((↑⟨((↑g₂).1, (↑(pG₁ ⊚ g)).2), ⋯⟩).2 e); All goals completed! 🐙 -- Hence G is isomorphic to prod2D by uniqueness of products sumDD:IrreflexiveGraphprod2D:IrreflexiveGraphj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂p₁:prod2D ⟶ IG2p₂:prod2D ⟶ Dh_prod2D:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gG:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩pG₁:G ⟶ IG2 := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩h_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDDh_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ IG2) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prod2D⊢ sumDD ≅ prod2D -- sumDD and prod2D are therefore isomorphic by transitivity All goals completed! 🐙

For part (b), we show that {D × D = D} (up to isomorphism).

open IrreflexiveGraph in noncomputable example {prodDD : IrreflexiveGraph} (p₁ : prodDD ⟶ D) (p₂ : prodDD ⟶ D) (h_prodDD : ∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ D) (g₂ : Y ⟶ D), (∃! g : Y ⟶ prodDD, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g)) : prodDD ≅ D := { hom := { val := ⟨p₁.val.1, fun _ ↦ ()⟩ property := prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ ((↑p₁).1, fun x ↦ ()).2 ⊚ prodDD.toSrc = D.toSrc ⊚ ((↑p₁).1, fun x ↦ ()).1 ∧ ((↑p₁).1, fun x ↦ ()).2 ⊚ prodDD.toTgt = D.toTgt ⊚ ((↑p₁).1, fun x ↦ ()).1 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ ((↑p₁).1, fun x ↦ ()).2 ⊚ prodDD.toSrc = D.toSrc ⊚ ((↑p₁).1, fun x ↦ ()).1prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ ((↑p₁).1, fun x ↦ ()).2 ⊚ prodDD.toTgt = D.toTgt ⊚ ((↑p₁).1, fun x ↦ ()).1 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ ((↑p₁).1, fun x ↦ ()).2 ⊚ prodDD.toSrc = D.toSrc ⊚ ((↑p₁).1, fun x ↦ ()).1prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ ((↑p₁).1, fun x ↦ ()).2 ⊚ prodDD.toTgt = D.toTgt ⊚ ((↑p₁).1, fun x ↦ ()).1 All goals completed! 🐙 } inv := Classical.choose (h_prodDD D (𝟙 D) (𝟙 D)).exists hom_inv_id := prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ = 𝟙 prodDD prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ = 𝟙 prodDD prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ ∧ p₂ = p₂ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ 𝟙 prodDD ∧ p₂ = p₂ ⊚ 𝟙 prodDD prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ ∧ p₂ = p₂ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₂ = p₂ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ All goals completed! 🐙 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₂ = p₂ ⊚ Classical.choose ⋯ ⊚ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₂ = ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ (↑p₂).1 = (↑⟨((↑p₁).1, fun x ↦ ()), ⋯⟩).1prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ (↑p₂).2 = (↑⟨((↑p₁).1, fun x ↦ ()), ⋯⟩).2 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ (↑p₂).1 = (↑⟨((↑p₁).1, fun x ↦ ()), ⋯⟩).1 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯x:prodDD.carrierA⊢ (↑p₂).1 x = (↑⟨((↑p₁).1, fun x ↦ ()), ⋯⟩).1 x All goals completed! 🐙 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ (↑p₂).2 = (↑⟨((↑p₁).1, fun x ↦ ()), ⋯⟩).2 All goals completed! 🐙 prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ 𝟙 prodDD ∧ p₂ = p₂ ⊚ 𝟙 prodDD prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ 𝟙 prodDDprodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₂ = p₂ ⊚ 𝟙 prodDD prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₁ = p₁ ⊚ 𝟙 prodDDprodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gh_finv_comm:𝟙 D = p₁ ⊚ Classical.choose ⋯ ∧ 𝟙 D = p₂ ⊚ Classical.choose ⋯⊢ p₂ = p₂ ⊚ 𝟙 prodDD All goals completed! 🐙 inv_hom_id := prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g⊢ ⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ ⊚ Classical.choose ⋯ = 𝟙 D prodDD:IrreflexiveGraphp₁:prodDD ⟶ Dp₂:prodDD ⟶ Dh_prodDD:∀ (Y : IrreflexiveGraph) (g₁ g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gx:D.carrierA⊢ (↑(⟨((↑p₁).1, fun x ↦ ()), ⋯⟩ ⊚ Classical.choose ⋯)).1 x = (↑(𝟙 D)).1 x All goals completed! 🐙 }

The proof of {A × D = D + D} in part (c) is very similar to the proof of {D + D = 2 \times D} in part (a).

open IrreflexiveGraph in noncomputable example {prodAD sumDD : IrreflexiveGraph} (p₁ : prodAD ⟶ A) (p₂ : prodAD ⟶ D) (h_prodAD : ∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), (∃! g : Y ⟶ prodAD, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ g)) (j₁ : D ⟶ sumDD) (j₂ : D ⟶ sumDD) (h_sumDD : ∀ (X : IrreflexiveGraph) (f₁ : D ⟶ X) (f₂ : D ⟶ X), (∃! f : sumDD ⟶ X, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂)) : prodAD ≅ sumDD := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂⊢ prodAD ≅ sumDD -- Define graph G isomorphic to both prodAD and sumDD prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ prodAD ≅ sumDD -- Define projections from G let pG₁ : G ⟶ A := { val := ⟨Empty.elim, fun x ↦ x⟩ property := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toSrc = A.toSrc ⊚ (Empty.elim, fun x ↦ x).1 ∧ (Empty.elim, fun x ↦ x).2 ⊚ G.toTgt = A.toTgt ⊚ (Empty.elim, fun x ↦ x).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toSrc = A.toSrc ⊚ (Empty.elim, fun x ↦ x).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toTgt = A.toTgt ⊚ (Empty.elim, fun x ↦ x).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toSrc = A.toSrc ⊚ (Empty.elim, fun x ↦ x).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }⊢ (Empty.elim, fun x ↦ x).2 ⊚ G.toTgt = A.toTgt ⊚ (Empty.elim, fun x ↦ x).1 (prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }x:G.carrierA⊢ ((Empty.elim, fun x ↦ x).2 ⊚ G.toTgt) x = (A.toTgt ⊚ (Empty.elim, fun x ↦ x).1) x; All goals completed! 🐙) } let pG₂ : G ⟶ D := { val := ⟨Empty.elim, fun x ↦ ()⟩ property := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toSrc = D.toSrc ⊚ (Empty.elim, fun x ↦ ()).1 ∧ (Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt = D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toSrc = D.toSrc ⊚ (Empty.elim, fun x ↦ ()).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt = D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toSrc = D.toSrc ⊚ (Empty.elim, fun x ↦ ()).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩⊢ (Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt = D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1 (prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩x:G.carrierA⊢ ((Empty.elim, fun x ↦ ()).2 ⊚ G.toTgt) x = (D.toTgt ⊚ (Empty.elim, fun x ↦ ()).1) x; All goals completed! 🐙) } -- Define injections to G let jG₁ : D ⟶ G := { val := ⟨Empty.elim, fun _ ↦ 0⟩ property := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 0).1 ∧ (Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 0).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 0).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩⊢ (Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1 (prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩x:D.carrierA⊢ ((Empty.elim, fun x ↦ 0).2 ⊚ D.toTgt) x = (G.toTgt ⊚ (Empty.elim, fun x ↦ 0).1) x; All goals completed! 🐙) } let jG₂ : D ⟶ G := { val := ⟨Empty.elim, fun _ ↦ 1⟩ property := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 1).1 ∧ (Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 1).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toSrc = G.toSrc ⊚ (Empty.elim, fun x ↦ 1).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩⊢ (Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt = G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1 (prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩x:D.carrierA⊢ ((Empty.elim, fun x ↦ 1).2 ⊚ D.toTgt) x = (G.toTgt ⊚ (Empty.elim, fun x ↦ 1).1) x; All goals completed! 🐙) } -- Show that G satisfies universal property for products have h_prodG : ∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), (∃! g : Y ⟶ G, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂⊢ prodAD ≅ sumDD prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g use { val := ⟨g₂.val.1, g₁.val.2⟩ property := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toSrc = G.toSrc ⊚ ((↑g₂).1, (↑g₁).2).1 ∧ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt = G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toSrc = G.toSrc ⊚ ((↑g₂).1, (↑g₁).2).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt = G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toSrc = G.toSrc ⊚ ((↑g₂).1, (↑g₁).2).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt = G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1 (prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ De:Y.carrierA⊢ (((↑g₂).1, (↑g₁).2).2 ⊚ Y.toTgt) e = (G.toTgt ⊚ ((↑g₂).1, (↑g₁).2).1) e; All goals completed! 🐙) } prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) ⟨((↑g₂).1, (↑g₁).2), ⋯⟩prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ∀ (y : Y ⟶ G), (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) y → y = ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ g₁ = pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ g₂ = pG₂ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ g₁ = pG₁ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ All goals completed! 🐙 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ g₂ = pG₂ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ De:Y.carrierA⊢ (↑g₂).1 e = (↑(pG₂ ⊚ ⟨((↑g₂).1, (↑g₁).2), ⋯⟩)).1 e All goals completed! 🐙 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ D⊢ ∀ (y : Y ⟶ G), (fun g ↦ g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g) y → y = ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ g⊢ g = ⟨((↑g₂).1, (↑g₁).2), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierA⊢ (↑g).1 e = (↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).1 eprodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierD⊢ ↑((↑g).2 e) = ↑((↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).2 e) prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierA⊢ (↑g).1 e = (↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).1 e All goals completed! 🐙 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierD⊢ ↑((↑g).2 e) = ↑((↑⟨((↑g₂).1, (↑g₁).2), ⋯⟩).2 e) prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩Y:IrreflexiveGraphg₁:Y ⟶ Ag₂:Y ⟶ Dg:Y ⟶ Ghg:g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ ge:Y.carrierD⊢ ↑((↑g).2 e) = ↑((↑⟨((↑g₂).1, (↑(pG₁ ⊚ g)).2), ⋯⟩).2 e); All goals completed! 🐙 -- Hence G is isomorphic to prodAD by uniqueness of products prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodAD⊢ prodAD ≅ sumDD -- Show that G satisfies universal property for sums have h_sumG : ∀ (X : IrreflexiveGraph) (f₁ : D ⟶ X) (f₂ : D ⟶ X), (∃! f : G ⟶ X, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂⊢ prodAD ≅ sumDD prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂ use { val := ⟨Empty.elim, fun | 0 => f₁.val.2 () | 1 => f₂.val.2 ()⟩ property := prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 ∧ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1 (prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:G.carrierA⊢ ((Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).2 ⊚ G.toTgt) x = (X.toTgt ⊚ (Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()).1) x; All goals completed! 🐙) } prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ ∀ (y : G ⟶ X), (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) y → y = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ f₁ = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₁prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ f₂ = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂ all_goals prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:D.carrierA⊢ (↑f₂).1 x = (↑(⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂)).1 xprodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:D.carrierD⊢ (↑f₂).2 x = (↑(⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂)).2 x prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xx:D.carrierD⊢ (↑f₂).2 x = (↑(⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ ⊚ jG₂)).2 x All goals completed! 🐙 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ X⊢ ∀ (y : G ⟶ X), (fun f ↦ f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂) y → y = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ f = ⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩ prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierA⊢ (↑f).1 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).1 xprodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierD⊢ (↑f).2 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 x prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierA⊢ (↑f).1 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).1 x All goals completed! 🐙 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂x:G.carrierD⊢ (↑f).2 x = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 x prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨0, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨0, ⋯⟩)prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨1, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨1, ⋯⟩) prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨0, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨0, ⋯⟩) prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨0, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑(f ⊚ jG₁)).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨0, ⋯⟩); All goals completed! 🐙 prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨1, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑f₂).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨1, ⋯⟩) prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADX:IrreflexiveGraphf₁:D ⟶ Xf₂:D ⟶ Xf:G ⟶ Xhf:f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂⊢ (↑f).2 ((fun i ↦ i) ⟨1, ⋯⟩) = (↑⟨(Empty.elim, fun x ↦ match x with | 0 => (↑f₁).2 () | 1 => (↑(f ⊚ jG₂)).2 ()), ⋯⟩).2 ((fun i ↦ i) ⟨1, ⋯⟩); All goals completed! 🐙 -- Hence G is isomorphic to sumDD by uniqueness of sums prodAD:IrreflexiveGraphsumDD:IrreflexiveGraphp₁:prodAD ⟶ Ap₂:prodAD ⟶ Dh_prodAD:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = p₁ ⊚ g ∧ g₂ = p₂ ⊚ gj₁:D ⟶ sumDDj₂:D ⟶ sumDDh_sumDD:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ j₁ ∧ f₂ = f ⊚ j₂G:IrreflexiveGraph := { carrierA := Empty, carrierD := Fin 2, toSrc := Empty.elim, toTgt := Empty.elim }pG₁:G ⟶ A := ⟨(Empty.elim, fun x ↦ x), ⋯⟩pG₂:G ⟶ D := ⟨(Empty.elim, fun x ↦ ()), ⋯⟩jG₁:D ⟶ G := ⟨(Empty.elim, fun x ↦ 0), ⋯⟩jG₂:D ⟶ G := ⟨(Empty.elim, fun x ↦ 1), ⋯⟩h_prodG:∀ (Y : IrreflexiveGraph) (g₁ : Y ⟶ A) (g₂ : Y ⟶ D), ∃! g, g₁ = pG₁ ⊚ g ∧ g₂ = pG₂ ⊚ gh_iso_prod:G ≅ prodADh_sumG:∀ (X : IrreflexiveGraph) (f₁ f₂ : D ⟶ X), ∃! f, f₁ = f ⊚ jG₁ ∧ f₂ = f ⊚ jG₂h_iso_sum:G ≅ sumDD⊢ prodAD ≅ sumDD -- prodAD and sumDD are therefore isomorphic by transitivity All goals completed! 🐙
Exercise 3 (p. 268)

Reread Section 5 of Session 15 and find a method, starting from presentations of X^{↻\alpha} and Y^{↻\beta}, to construct presentations of

(a) {X^{↻\alpha} + Y^{↻\beta}}

(b) {X^{↻\alpha} \times Y^{↻\beta}}

Part (b) is harder than part (a).

Solution: Exercise 3

(a) Given a presentation of X^{↻\alpha} with a list of generators (\mathbf{L}_X) and a list of relations (\mathbf{R}_X), and a presentation of Y^{↻\beta} with a list of generators (\mathbf{L}_Y) and a list of relations (\mathbf{R}_Y), then a presentation of {X^{↻\alpha} + Y^{↻\beta}} can be constructed simply from the disjoint union of generators (\mathbf{L}_X) and (\mathbf{L}_Y), together with the disjoint union of relations (\mathbf{R}_X) and (\mathbf{R}_Y) (with suitable relabelling of the generators and relations to avoid name clashes between generators injected from X^{↻\alpha} and those injected from Y^{↻\beta}).

TODO Exercise 24.3 (b)