Session 22: Universal mapping properties; Incidence relations🔗

4. Basic figure-types, singular figures, and incidence, in the category of graphs🔗

Exercise 1 (p. 250)

Consider the diagram of graphs B_1 \leftarrow P \rightarrow B_2 Suppose it satisfies the fragment of the definition of product in which we test against only the two figure-types {X = A} and {X = D}. Prove that the diagram actually is a product, i.e. that the product property holds for all graphs X.

Solution: Exercise 1

We begin by creating helpers for two key equivalences.

open IrreflexiveGraph in
-- Establish equivalence between Hom(A, X) and arrows of X
def homAEquiv (X : IrreflexiveGraph) : (A ⟶ X) ≃ X.carrierA where
  toFun fAX := fAX.val.1 ()
  invFun xA := ⟨(fun _ ↦ xA, (fun
      | 0 => X.toSrc xA
      | 1 => X.toTgt xA
      : Fin 2 → X.carrierD)),
    byX:IrreflexiveGraphxA:X.carrierA⊢ (fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA).2 ⊚
      A.toSrc =
    X.toSrc ⊚
      (fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA).1 ∧
  (fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA).2 ⊚
      A.toTgt =
    X.toTgt ⊚
      (fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA).1
      constructorleftX:IrreflexiveGraphxA:X.carrierA⊢ (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).2 ⊚
    A.toSrc =
  X.toSrc ⊚
    (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).1rightX:IrreflexiveGraphxA:X.carrierA⊢ (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).2 ⊚
    A.toTgt =
  X.toTgt ⊚
    (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).1 <;>leftX:IrreflexiveGraphxA:X.carrierA⊢ (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).2 ⊚
    A.toSrc =
  X.toSrc ⊚
    (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).1rightX:IrreflexiveGraphxA:X.carrierA⊢ (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).2 ⊚
    A.toTgt =
  X.toTgt ⊚
    (fun x ↦ xA, fun x ↦
        match x with
        | 0 => X.toSrc xA
        | 1 => X.toTgt xA).1 (funextright.hX:IrreflexiveGraphxA:X.carrierAx✝:A.carrierA⊢ ((fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA).2 ⊚
      A.toTgt)
    x✝ =
  (X.toTgt ⊚
      (fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA).1)
    x✝; rflAll goals completed! 🐙)
  ⟩
  left_inv fAX := byX:IrreflexiveGraphfAX:A ⟶ X⊢ (fun xA ↦
      ⟨(fun x ↦ xA, fun x ↦
          match x with
          | 0 => X.toSrc xA
          | 1 => X.toTgt xA),
        ⋯⟩)
    ((fun fAX ↦ (↑fAX).1 ()) fAX) =
  fAX
    apply IrreflexiveGraph.hom_exthAX:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
          ⟨(fun x ↦ xA, fun x ↦
              match x with
              | 0 => X.toSrc xA
              | 1 => X.toTgt xA),
            ⋯⟩)
        ((fun fAX ↦ (↑fAX).1 ()) fAX))).1 =
  (↑fAX).1hDX:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
          ⟨(fun x ↦ xA, fun x ↦
              match x with
              | 0 => X.toSrc xA
              | 1 => X.toTgt xA),
            ⋯⟩)
        ((fun fAX ↦ (↑fAX).1 ()) fAX))).2 =
  (↑fAX).2
    ·hAX:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
          ⟨(fun x ↦ xA, fun x ↦
              match x with
              | 0 => X.toSrc xA
              | 1 => X.toTgt xA),
            ⋯⟩)
        ((fun fAX ↦ (↑fAX).1 ()) fAX))).1 =
  (↑fAX).1 funexthA.hX:IrreflexiveGraphfAX:A ⟶ Xx✝:A.carrierA⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).1
    x✝ =
  (↑fAX).1 x✝; rflAll goals completed! 🐙
    ·hDX:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
          ⟨(fun x ↦ xA, fun x ↦
              match x with
              | 0 => X.toSrc xA
              | 1 => X.toTgt xA),
            ⋯⟩)
        ((fun fAX ↦ (↑fAX).1 ()) fAX))).2 =
  (↑fAX).2 funext ihD.hX:IrreflexiveGraphfAX:A ⟶ Xi:A.carrierD⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).2
    i =
  (↑fAX).2 i
      change Fin 2 at ihD.hX:IrreflexiveGraphfAX:A ⟶ Xi:Fin 2⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).2
    i =
  (↑fAX).2 i
      fin_cases ihD.h.«0»X:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).2
    ((fun i ↦ i) ⟨0, ⋯⟩) =
  (↑fAX).2 ((fun i ↦ i) ⟨0, ⋯⟩)hD.h.«1»X:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).2
    ((fun i ↦ i) ⟨1, ⋯⟩) =
  (↑fAX).2 ((fun i ↦ i) ⟨1, ⋯⟩)
      ·hD.h.«0»X:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).2
    ((fun i ↦ i) ⟨0, ⋯⟩) =
  (↑fAX).2 ((fun i ↦ i) ⟨0, ⋯⟩) exact (congr_fun fAX.property.1 ()).symmAll goals completed! 🐙
      ·hD.h.«1»X:IrreflexiveGraphfAX:A ⟶ X⊢ (↑((fun xA ↦
            ⟨(fun x ↦ xA, fun x ↦
                match x with
                | 0 => X.toSrc xA
                | 1 => X.toTgt xA),
              ⋯⟩)
          ((fun fAX ↦ (↑fAX).1 ()) fAX))).2
    ((fun i ↦ i) ⟨1, ⋯⟩) =
  (↑fAX).2 ((fun i ↦ i) ⟨1, ⋯⟩) exact (congr_fun fAX.property.2 ()).symmAll goals completed! 🐙
  right_inv xA := rfl

open IrreflexiveGraph in
-- Establish equivalence between Hom(D, X) and dots of X
def homDEquiv (X : IrreflexiveGraph) : (D ⟶ X) ≃ X.carrierD where
  toFun fDX := fDX.val.2 ()
  invFun xD := ⟨(Empty.elim, fun _ ↦ xD),
    byX:IrreflexiveGraphxD:X.carrierD⊢ (Empty.elim, fun x ↦ xD).2 ⊚ D.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ xD).1 ∧
  (Empty.elim, fun x ↦ xD).2 ⊚ D.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ xD).1
      constructorleftX:IrreflexiveGraphxD:X.carrierD⊢ (Empty.elim, fun x ↦ xD).2 ⊚ D.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ xD).1rightX:IrreflexiveGraphxD:X.carrierD⊢ (Empty.elim, fun x ↦ xD).2 ⊚ D.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ xD).1 <;>leftX:IrreflexiveGraphxD:X.carrierD⊢ (Empty.elim, fun x ↦ xD).2 ⊚ D.toSrc = X.toSrc ⊚ (Empty.elim, fun x ↦ xD).1rightX:IrreflexiveGraphxD:X.carrierD⊢ (Empty.elim, fun x ↦ xD).2 ⊚ D.toTgt = X.toTgt ⊚ (Empty.elim, fun x ↦ xD).1 (funext eright.hX:IrreflexiveGraphxD:X.carrierDe:D.carrierA⊢ ((Empty.elim, fun x ↦ xD).2 ⊚ D.toTgt) e = (X.toTgt ⊚ (Empty.elim, fun x ↦ xD).1) e; exact Empty.elim eAll goals completed! 🐙)
  ⟩
  left_inv fDX := byX:IrreflexiveGraphfDX:D ⟶ X⊢ (fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX) = fDX
    apply IrreflexiveGraph.hom_exthAX:IrreflexiveGraphfDX:D ⟶ X⊢ (↑((fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX))).1 = (↑fDX).1hDX:IrreflexiveGraphfDX:D ⟶ X⊢ (↑((fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX))).2 = (↑fDX).2
    ·hAX:IrreflexiveGraphfDX:D ⟶ X⊢ (↑((fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX))).1 = (↑fDX).1 funext ehA.hX:IrreflexiveGraphfDX:D ⟶ Xe:D.carrierA⊢ (↑((fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX))).1 e = (↑fDX).1 e; exact Empty.elim eAll goals completed! 🐙
    ·hDX:IrreflexiveGraphfDX:D ⟶ X⊢ (↑((fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX))).2 = (↑fDX).2 funexthD.hX:IrreflexiveGraphfDX:D ⟶ Xx✝:D.carrierD⊢ (↑((fun xD ↦ ⟨(Empty.elim, fun x ↦ xD), ⋯⟩) ((fun fDX ↦ (↑fDX).2 ()) fDX))).2 x✝ = (↑fDX).2 x✝; rflAll goals completed! 🐙
  right_inv xD := rfl

Our proof then follows.

open IrreflexiveGraph in
example
    (P B₁ B₂ : IrreflexiveGraph) (p₁ : P ⟶ B₁) (p₂ : P ⟶ B₂)
    -- Satisfies fragment of definition of product for X = A
    (hA : ∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂),
        ∃! fAP : A ⟶ P, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂)
    -- Satisfies fragment of definition of product for X = D
    (hD : ∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂),
        ∃! fDP : D ⟶ P, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂)
    : ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂),
        ∃! f : X ⟶ P, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂ := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
  intro X f₁P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁⊢ ∀ (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂ f₂P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂

  -- For each arrow in X, get equivalent morphism fAX : A ⟶ X
  let fAX (xA : X.carrierA) : A ⟶ X := (homAEquiv X).symm xAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xA⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
  -- For each dot in X, get equivalent morphism fDX : D ⟶ X
  let fDX (xD : X.carrierD) : D ⟶ X := (homDEquiv X).symm xDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xD⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
  -- Construct fA : X.carrierA ⟶ P.carrierA
  let fA (xA : X.carrierA) : P.carrierA :=
    -- Compose fAX with pair f₁, f₂ to obtain A ⟶ B₁, A ⟶ B₂
    let h_uniq := hA (f₁ ⊚ fAX xA) (f₂ ⊚ fAX xA)
    -- which we then use to find unique arrow in P
    (homAEquiv P) (Classical.choose h_uniq)P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
  -- Construct fD : X.carrierD ⟶ P.carrierD (cf. fA)
  let fD (xD : X.carrierD) : P.carrierD :=
    let h_uniq := hD (f₁ ⊚ fDX xD) (f₂ ⊚ fDX xD)
    (homDEquiv P) (Classical.choose h_uniq)P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
  -- For each arrow in P, get morphism fAP : A ⟶ P
  let fAP (pA : P.carrierA) : A ⟶ P := (homAEquiv P).symm pAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pA⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
  -- For each dot in P, get morphism fDP : D ⟶ P
  let fDP (pD : P.carrierD) : D ⟶ P := (homDEquiv P).symm pDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pD⊢ ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂

  -- Show that p₁ ⊚ fAP = f₁ ⊚ fAX ∧ p₂ ⊚ fAP = f₂ ⊚ fAX
  have hA_proj : ∀ xA : X.carrierA,
      p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧
      p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xA := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
    intro xAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDxA:X.carrierA⊢ p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xA
    dsimp [fAX, fAP]P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDxA:X.carrierA⊢ p₁ ⊚ (homAEquiv P).symm (fA xA) = f₁ ⊚ (homAEquiv X).symm xA ∧
  p₂ ⊚ (homAEquiv P).symm (fA xA) = f₂ ⊚ (homAEquiv X).symm xA
    rw [Equiv.symm_apply_apply]P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDxA:X.carrierA⊢ p₁ ⊚ Classical.choose ⋯ = f₁ ⊚ (homAEquiv X).symm xA ∧ p₂ ⊚ Classical.choose ⋯ = f₂ ⊚ (homAEquiv X).symm xA
    exact (Classical.choose_spec (hA _ _)).1All goals completed! 🐙
  -- Show that p₁ ⊚ fDP = f₁ ⊚ fDX ∧ p₂ ⊚ fDP = f₂ ⊚ fDX
  have hD_proj : ∀ xD : X.carrierD,
      p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧
      p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xD := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
    intro xDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAxD:X.carrierD⊢ p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xD
    dsimp [fDX, fDP]P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAxD:X.carrierD⊢ p₁ ⊚ (homDEquiv P).symm (fD xD) = f₁ ⊚ (homDEquiv X).symm xD ∧
  p₂ ⊚ (homDEquiv P).symm (fD xD) = f₂ ⊚ (homDEquiv X).symm xD
    rw [Equiv.symm_apply_apply]P:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAxD:X.carrierD⊢ p₁ ⊚ Classical.choose ⋯ = f₁ ⊚ (homDEquiv X).symm xD ∧ p₂ ⊚ Classical.choose ⋯ = f₂ ⊚ (homDEquiv X).symm xD
    exact (Classical.choose_spec (hD _ _)).1All goals completed! 🐙

  -- Show that P.toSrc ⊚ fA = fD ⊚ X.toSrc
  have hSrc_comm :
      ∀ xA : X.carrierA, P.toSrc (fA xA) = fD (X.toSrc xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
    intro xAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierA⊢ P.toSrc (fA xA) = fD (X.toSrc xA)
    -- Show that p₁ ⊚ fDP = f₁ ⊚ fDX
    have hp₁ : p₁ ⊚ fDP (P.toSrc (fA xA)) =
               f₁ ⊚ fDX (X.toSrc xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
      -- Transform goal from map equality to dot equality
      apply (homDEquiv B₁).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierA⊢ (homDEquiv B₁) (p₁ ⊚ fDP (P.toSrc (fA xA))) = (homDEquiv B₁) (f₁ ⊚ fDX (X.toSrc xA))
      calc (p₁.val.2 ⊚ P.toSrc) (fA xA)
        _ = (B₁.toSrc ⊚ p₁.val.1) (fA xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierA⊢ ((↑p₁).2 ⊚ P.toSrc) (fA xA) = (B₁.toSrc ⊚ (↑p₁).1) (fA xA) rw [p₁.property.1]All goals completed! 🐙
        _ = B₁.toSrc ((p₁ ⊚ fAP (fA xA)).val.1 ())
                                           := rfl
        _ = B₁.toSrc ((f₁ ⊚ fAX xA).val.1 ())
                                           := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierA⊢ B₁.toSrc ((↑(p₁ ⊚ fAP (fA xA))).1 ()) = B₁.toSrc ((↑(f₁ ⊚ fAX xA)).1 ()) rw [(hA_proj xA).1]All goals completed! 🐙
        _ = (B₁.toSrc ⊚ f₁.val.1) xA      := rfl
        _ = (f₁.val.2 ⊚ X.toSrc) xA       := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierA⊢ (B₁.toSrc ⊚ (↑f₁).1) xA = ((↑f₁).2 ⊚ X.toSrc) xA rw [f₁.property.1]All goals completed! 🐙
    -- Show that p₂ ⊚ fDP = f₂ ⊚ fDX
    have hp₂ : p₂ ⊚ fDP (P.toSrc (fA xA)) =
               f₂ ⊚ fDX (X.toSrc xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
      -- Transform goal from map equality to dot equality
      apply (homDEquiv B₂).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)⊢ (homDEquiv B₂) (p₂ ⊚ fDP (P.toSrc (fA xA))) = (homDEquiv B₂) (f₂ ⊚ fDX (X.toSrc xA))
      calc (p₂.val.2 ⊚ P.toSrc) (fA xA)
        _ = (B₂.toSrc ⊚ p₂.val.1) (fA xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)⊢ ((↑p₂).2 ⊚ P.toSrc) (fA xA) = (B₂.toSrc ⊚ (↑p₂).1) (fA xA) rw [p₂.property.1]All goals completed! 🐙
        _ = B₂.toSrc ((p₂ ⊚ fAP (fA xA)).val.1 ())
                                           := rfl
        _ = B₂.toSrc ((f₂ ⊚ fAX xA).val.1 ())
                                           := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)⊢ B₂.toSrc ((↑(p₂ ⊚ fAP (fA xA))).1 ()) = B₂.toSrc ((↑(f₂ ⊚ fAX xA)).1 ()) rw [(hA_proj xA).2]All goals completed! 🐙
        _ = (B₂.toSrc ⊚ f₂.val.1) xA      := rfl
        _ = (f₂.val.2 ⊚ X.toSrc) xA       := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)⊢ (B₂.toSrc ⊚ (↑f₂).1) xA = ((↑f₂).2 ⊚ X.toSrc) xA rw [f₂.property.1]All goals completed! 🐙
    -- Transform goal from dot equality to map equality
    apply (homDEquiv P).symm.injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)hp₂:p₂ ⊚ fDP (P.toSrc (fA xA)) = f₂ ⊚ fDX (X.toSrc xA)⊢ (homDEquiv P).symm (P.toSrc (fA xA)) = (homDEquiv P).symm (fD (X.toSrc xA))
    change fDP (P.toSrc (fA xA)) = fDP (fD (X.toSrc xA))aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)hp₂:p₂ ⊚ fDP (P.toSrc (fA xA)) = f₂ ⊚ fDX (X.toSrc xA)⊢ fDP (P.toSrc (fA xA)) = fDP (fD (X.toSrc xA))
    -- Get unique witness for D ⟶ P
    have h_uniq := (Classical.choose_spec (hD (f₁ ⊚ fDX (X.toSrc xA))
        (f₂ ⊚ fDX (X.toSrc xA)))).2aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)hp₂:p₂ ⊚ fDP (P.toSrc (fA xA)) = f₂ ⊚ fDX (X.toSrc xA)h_uniq:∀ (y : D ⟶ P),
  (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX (X.toSrc xA) ∧ p₂ ⊚ fDP = f₂ ⊚ fDX (X.toSrc xA)) y → y = Classical.choose ⋯⊢ fDP (P.toSrc (fA xA)) = fDP (fD (X.toSrc xA))
    -- Show that lhs and rhs both equal unique witness
    have h_left := h_uniq (fDP (P.toSrc (fA xA))) ⟨hp₁, hp₂⟩aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)hp₂:p₂ ⊚ fDP (P.toSrc (fA xA)) = f₂ ⊚ fDX (X.toSrc xA)h_uniq:∀ (y : D ⟶ P),
  (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX (X.toSrc xA) ∧ p₂ ⊚ fDP = f₂ ⊚ fDX (X.toSrc xA)) y → y = Classical.choose ⋯h_left:fDP (P.toSrc (fA xA)) = Classical.choose ⋯⊢ fDP (P.toSrc (fA xA)) = fDP (fD (X.toSrc xA))
    have h_right := h_uniq (fDP (fD (X.toSrc xA)))
        (hD_proj (X.toSrc xA))aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDxA:X.carrierAhp₁:p₁ ⊚ fDP (P.toSrc (fA xA)) = f₁ ⊚ fDX (X.toSrc xA)hp₂:p₂ ⊚ fDP (P.toSrc (fA xA)) = f₂ ⊚ fDX (X.toSrc xA)h_uniq:∀ (y : D ⟶ P),
  (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX (X.toSrc xA) ∧ p₂ ⊚ fDP = f₂ ⊚ fDX (X.toSrc xA)) y → y = Classical.choose ⋯h_left:fDP (P.toSrc (fA xA)) = Classical.choose ⋯h_right:fDP (fD (X.toSrc xA)) = Classical.choose ⋯⊢ fDP (P.toSrc (fA xA)) = fDP (fD (X.toSrc xA))
    rw [h_left, h_right]All goals completed! 🐙

  -- Show that P.toTgt ⊚ fA = fD ⊚ X.toTgt (cf. hSrc_comm)
  have hTgt_comm :
      ∀ xA : X.carrierA, P.toTgt (fA xA) = fD (X.toTgt xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
    intro xAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierA⊢ P.toTgt (fA xA) = fD (X.toTgt xA)
    have hp₁ : p₁ ⊚ fDP (P.toTgt (fA xA)) =
               f₁ ⊚ fDX (X.toTgt xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
      apply (homDEquiv B₁).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierA⊢ (homDEquiv B₁) (p₁ ⊚ fDP (P.toTgt (fA xA))) = (homDEquiv B₁) (f₁ ⊚ fDX (X.toTgt xA))
      calc (p₁.val.2 ⊚ P.toTgt) (fA xA)
        _ = (B₁.toTgt ⊚ p₁.val.1) (fA xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierA⊢ ((↑p₁).2 ⊚ P.toTgt) (fA xA) = (B₁.toTgt ⊚ (↑p₁).1) (fA xA) rw [p₁.property.2]All goals completed! 🐙
        _ = B₁.toTgt ((p₁ ⊚ fAP (fA xA)).val.1 ())
                                           := rfl
        _ = B₁.toTgt ((f₁ ⊚ fAX xA).val.1 ())
                                           := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierA⊢ B₁.toTgt ((↑(p₁ ⊚ fAP (fA xA))).1 ()) = B₁.toTgt ((↑(f₁ ⊚ fAX xA)).1 ()) rw [(hA_proj xA).1]All goals completed! 🐙
        _ = (B₁.toTgt ⊚ f₁.val.1) xA      := rfl
        _ = (f₁.val.2 ⊚ X.toTgt) xA       := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierA⊢ (B₁.toTgt ⊚ (↑f₁).1) xA = ((↑f₁).2 ⊚ X.toTgt) xA rw [f₁.property.2]All goals completed! 🐙
    have hp₂ : p₂ ⊚ fDP (P.toTgt (fA xA)) =
               f₂ ⊚ fDX (X.toTgt xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
      apply (homDEquiv B₂).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)⊢ (homDEquiv B₂) (p₂ ⊚ fDP (P.toTgt (fA xA))) = (homDEquiv B₂) (f₂ ⊚ fDX (X.toTgt xA))
      calc (p₂.val.2 ⊚ P.toTgt) (fA xA)
        _ = (B₂.toTgt ⊚ p₂.val.1) (fA xA) := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)⊢ ((↑p₂).2 ⊚ P.toTgt) (fA xA) = (B₂.toTgt ⊚ (↑p₂).1) (fA xA) rw [p₂.property.2]All goals completed! 🐙
        _ = B₂.toTgt ((p₂ ⊚ fAP (fA xA)).val.1 ())
                                           := rfl
        _ = B₂.toTgt ((f₂ ⊚ fAX xA).val.1 ())
                                           := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)⊢ B₂.toTgt ((↑(p₂ ⊚ fAP (fA xA))).1 ()) = B₂.toTgt ((↑(f₂ ⊚ fAX xA)).1 ()) rw [(hA_proj xA).2]All goals completed! 🐙
        _ = (B₂.toTgt ⊚ f₂.val.1) xA      := rfl
        _ = (f₂.val.2 ⊚ X.toTgt) xA       := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)⊢ (B₂.toTgt ⊚ (↑f₂).1) xA = ((↑f₂).2 ⊚ X.toTgt) xA rw [f₂.property.2]All goals completed! 🐙
    apply (homDEquiv P).symm.injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)hp₂:p₂ ⊚ fDP (P.toTgt (fA xA)) = f₂ ⊚ fDX (X.toTgt xA)⊢ (homDEquiv P).symm (P.toTgt (fA xA)) = (homDEquiv P).symm (fD (X.toTgt xA))
    change fDP (P.toTgt (fA xA)) = fDP (fD (X.toTgt xA))aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)hp₂:p₂ ⊚ fDP (P.toTgt (fA xA)) = f₂ ⊚ fDX (X.toTgt xA)⊢ fDP (P.toTgt (fA xA)) = fDP (fD (X.toTgt xA))
    have h_uniq := (Classical.choose_spec (hD (f₁ ⊚ fDX (X.toTgt xA))
        (f₂ ⊚ fDX (X.toTgt xA)))).2aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)hp₂:p₂ ⊚ fDP (P.toTgt (fA xA)) = f₂ ⊚ fDX (X.toTgt xA)h_uniq:∀ (y : D ⟶ P),
  (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX (X.toTgt xA) ∧ p₂ ⊚ fDP = f₂ ⊚ fDX (X.toTgt xA)) y → y = Classical.choose ⋯⊢ fDP (P.toTgt (fA xA)) = fDP (fD (X.toTgt xA))
    have h_left := h_uniq (fDP (P.toTgt (fA xA))) ⟨hp₁, hp₂⟩aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)hp₂:p₂ ⊚ fDP (P.toTgt (fA xA)) = f₂ ⊚ fDX (X.toTgt xA)h_uniq:∀ (y : D ⟶ P),
  (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX (X.toTgt xA) ∧ p₂ ⊚ fDP = f₂ ⊚ fDX (X.toTgt xA)) y → y = Classical.choose ⋯h_left:fDP (P.toTgt (fA xA)) = Classical.choose ⋯⊢ fDP (P.toTgt (fA xA)) = fDP (fD (X.toTgt xA))
    have h_right := h_uniq (fDP (fD (X.toTgt xA)))
        (hD_proj (X.toTgt xA))aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)xA:X.carrierAhp₁:p₁ ⊚ fDP (P.toTgt (fA xA)) = f₁ ⊚ fDX (X.toTgt xA)hp₂:p₂ ⊚ fDP (P.toTgt (fA xA)) = f₂ ⊚ fDX (X.toTgt xA)h_uniq:∀ (y : D ⟶ P),
  (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX (X.toTgt xA) ∧ p₂ ⊚ fDP = f₂ ⊚ fDX (X.toTgt xA)) y → y = Classical.choose ⋯h_left:fDP (P.toTgt (fA xA)) = Classical.choose ⋯h_right:fDP (fD (X.toTgt xA)) = Classical.choose ⋯⊢ fDP (P.toTgt (fA xA)) = fDP (fD (X.toTgt xA))
    rw [h_left, h_right]All goals completed! 🐙

  -- Bundle fA and fD into morphism f : X ⟶ P
  let f : X ⟶ P := ⟨
    (fA, fD),
    byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)⊢ (fA, fD).2 ⊚ X.toSrc = P.toSrc ⊚ (fA, fD).1 ∧ (fA, fD).2 ⊚ X.toTgt = P.toTgt ⊚ (fA, fD).1
      constructorleftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)⊢ (fA, fD).2 ⊚ X.toSrc = P.toSrc ⊚ (fA, fD).1rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)⊢ (fA, fD).2 ⊚ X.toTgt = P.toTgt ⊚ (fA, fD).1 <;>leftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)⊢ (fA, fD).2 ⊚ X.toSrc = P.toSrc ⊚ (fA, fD).1rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)⊢ (fA, fD).2 ⊚ X.toTgt = P.toTgt ⊚ (fA, fD).1 funext xright.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)x:X.carrierA⊢ ((fA, fD).2 ⊚ X.toTgt) x = (P.toTgt ⊚ (fA, fD).1) x
      ·left.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)x:X.carrierA⊢ ((fA, fD).2 ⊚ X.toSrc) x = (P.toSrc ⊚ (fA, fD).1) x exact (hSrc_comm x).symmAll goals completed! 🐙
      ·right.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)x:X.carrierA⊢ ((fA, fD).2 ⊚ X.toTgt) x = (P.toTgt ⊚ (fA, fD).1) x exact (hTgt_comm x).symmAll goals completed! 🐙
  ⟩

  -- Show that f satisfies commutativity and uniqueness conditions
  use fhP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (fun f ↦ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂) f ∧ ∀ (y : X ⟶ P), (fun f ↦ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂) y → y = f
  constructorh.leftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (fun f ↦ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂) fh.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ ∀ (y : X ⟶ P), (fun f ↦ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂) y → y = f <;>h.leftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (fun f ↦ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂) fh.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ ∀ (y : X ⟶ P), (fun f ↦ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂) y → y = f dsimph.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ ∀ (y : X ⟶ P), p₁ ⊚ y = f₁ ∧ p₂ ⊚ y = f₂ → y = f
  -- Show that the triangles commute
  ·h.leftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂ constructorh.left.leftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ p₁ ⊚ f = f₁h.left.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ p₂ ⊚ f = f₂
    -- Show that p₁ ⊚ f = f₁
    ·h.left.leftP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ p₁ ⊚ f = f₁ apply IrreflexiveGraph.hom_exth.left.left.hAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₁ ⊚ f)).1 = (↑f₁).1h.left.left.hDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₁ ⊚ f)).2 = (↑f₁).2
      ·h.left.left.hAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₁ ⊚ f)).1 = (↑f₁).1 funext xAh.left.left.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xA:X.carrierA⊢ (↑(p₁ ⊚ f)).1 xA = (↑f₁).1 xA
        have h := (hA_proj xA).1h.left.left.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xA:X.carrierAh:p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA⊢ (↑(p₁ ⊚ f)).1 xA = (↑f₁).1 xA
        apply_fun (fun k => k.val.1 ()) at hh.left.left.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xA:X.carrierAh:(↑(p₁ ⊚ fAP (fA xA))).1 () = (↑(f₁ ⊚ fAX xA)).1 ()⊢ (↑(p₁ ⊚ f)).1 xA = (↑f₁).1 xA
        exact hAll goals completed! 🐙
      ·h.left.left.hDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₁ ⊚ f)).2 = (↑f₁).2 funext xDh.left.left.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xD:X.carrierD⊢ (↑(p₁ ⊚ f)).2 xD = (↑f₁).2 xD
        have h := (hD_proj xD).1h.left.left.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xD:X.carrierDh:p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD⊢ (↑(p₁ ⊚ f)).2 xD = (↑f₁).2 xD
        apply_fun (fun k => k.val.2 ()) at hh.left.left.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xD:X.carrierDh:(↑(p₁ ⊚ fDP (fD xD))).2 () = (↑(f₁ ⊚ fDX xD)).2 ()⊢ (↑(p₁ ⊚ f)).2 xD = (↑f₁).2 xD
        exact hAll goals completed! 🐙
    -- Show that p₂ ⊚ f = f₂
    ·h.left.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ p₂ ⊚ f = f₂ apply IrreflexiveGraph.hom_exth.left.right.hAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₂ ⊚ f)).1 = (↑f₂).1h.left.right.hDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₂ ⊚ f)).2 = (↑f₂).2
      ·h.left.right.hAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₂ ⊚ f)).1 = (↑f₂).1 funext xAh.left.right.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xA:X.carrierA⊢ (↑(p₂ ⊚ f)).1 xA = (↑f₂).1 xA
        have h := (hA_proj xA).2h.left.right.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xA:X.carrierAh:p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xA⊢ (↑(p₂ ⊚ f)).1 xA = (↑f₂).1 xA
        apply_fun (fun k => k.val.1 ()) at hh.left.right.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xA:X.carrierAh:(↑(p₂ ⊚ fAP (fA xA))).1 () = (↑(f₂ ⊚ fAX xA)).1 ()⊢ (↑(p₂ ⊚ f)).1 xA = (↑f₂).1 xA
        exact hAll goals completed! 🐙
      ·h.left.right.hDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ (↑(p₂ ⊚ f)).2 = (↑f₂).2 funext xDh.left.right.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xD:X.carrierD⊢ (↑(p₂ ⊚ f)).2 xD = (↑f₂).2 xD
        have h := (hD_proj xD).2h.left.right.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xD:X.carrierDh:p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xD⊢ (↑(p₂ ⊚ f)).2 xD = (↑f₂).2 xD
        apply_fun (fun k => k.val.2 ()) at hh.left.right.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩xD:X.carrierDh:(↑(p₂ ⊚ fDP (fD xD))).2 () = (↑(f₂ ⊚ fDX xD)).2 ()⊢ (↑(p₂ ⊚ f)).2 xD = (↑f₂).2 xD
        exact hAll goals completed! 🐙
  -- Prove uniqueness of f
  ·h.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩⊢ ∀ (y : X ⟶ P), p₁ ⊚ y = f₁ ∧ p₂ ⊚ y = f₂ → y = f intro f' ⟨hf'₁, hf'₂⟩h.rightP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂⊢ f' = f
    apply IrreflexiveGraph.hom_exth.right.hAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂⊢ (↑f').1 = (↑f).1h.right.hDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂⊢ (↑f').2 = (↑f).2
    -- Show that arrow maps are equal
    ·h.right.hAP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂⊢ (↑f').1 = (↑f).1 funext xAh.right.hA.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierA⊢ (↑f').1 xA = (↑f).1 xA
      have hp₁ : p₁ ⊚ fAP (f'.val.1 xA) = f₁ ⊚ fAX xA := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
        apply (homAEquiv B₁).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierA⊢ (homAEquiv B₁) (p₁ ⊚ fAP ((↑f').1 xA)) = (homAEquiv B₁) (f₁ ⊚ fAX xA)
        apply_fun (fun k => k.val.1 xA) at hf'₁aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhf'₁:(↑(p₁ ⊚ f')).1 xA = (↑f₁).1 xA⊢ (homAEquiv B₁) (p₁ ⊚ fAP ((↑f').1 xA)) = (homAEquiv B₁) (f₁ ⊚ fAX xA)
        exact hf'₁All goals completed! 🐙
      have hp₂ : p₂ ⊚ fAP (f'.val.1 xA) = f₂ ⊚ fAX xA := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
        apply (homAEquiv B₂).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xA⊢ (homAEquiv B₂) (p₂ ⊚ fAP ((↑f').1 xA)) = (homAEquiv B₂) (f₂ ⊚ fAX xA)
        apply_fun (fun k => k.val.1 xA) at hf'₂aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xAhf'₂:(↑(p₂ ⊚ f')).1 xA = (↑f₂).1 xA⊢ (homAEquiv B₂) (p₂ ⊚ fAP ((↑f').1 xA)) = (homAEquiv B₂) (f₂ ⊚ fAX xA)
        exact hf'₂All goals completed! 🐙
      apply (homAEquiv P).symm.injectiveh.right.hA.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xAhp₂:p₂ ⊚ fAP ((↑f').1 xA) = f₂ ⊚ fAX xA⊢ (homAEquiv P).symm ((↑f').1 xA) = (homAEquiv P).symm ((↑f).1 xA)
      change fAP (f'.val.1 xA) = fAP (fA xA)h.right.hA.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xAhp₂:p₂ ⊚ fAP ((↑f').1 xA) = f₂ ⊚ fAX xA⊢ fAP ((↑f').1 xA) = fAP (fA xA)
      have h_uniq := (Classical.choose_spec (hA (f₁ ⊚ fAX xA)
          (f₂ ⊚ fAX xA))).2h.right.hA.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xAhp₂:p₂ ⊚ fAP ((↑f').1 xA) = f₂ ⊚ fAX xAh_uniq:∀ (y : A ⟶ P), (fun fAP ↦ p₁ ⊚ fAP = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP = f₂ ⊚ fAX xA) y → y = Classical.choose ⋯⊢ fAP ((↑f').1 xA) = fAP (fA xA)
      have h_left := h_uniq (fAP (f'.val.1 xA)) ⟨hp₁, hp₂⟩h.right.hA.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xAhp₂:p₂ ⊚ fAP ((↑f').1 xA) = f₂ ⊚ fAX xAh_uniq:∀ (y : A ⟶ P), (fun fAP ↦ p₁ ⊚ fAP = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP = f₂ ⊚ fAX xA) y → y = Classical.choose ⋯h_left:fAP ((↑f').1 xA) = Classical.choose ⋯⊢ fAP ((↑f').1 xA) = fAP (fA xA)
      have h_right := h_uniq (fAP (fA xA)) (hA_proj xA)h.right.hA.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xA:X.carrierAhp₁:p₁ ⊚ fAP ((↑f').1 xA) = f₁ ⊚ fAX xAhp₂:p₂ ⊚ fAP ((↑f').1 xA) = f₂ ⊚ fAX xAh_uniq:∀ (y : A ⟶ P), (fun fAP ↦ p₁ ⊚ fAP = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP = f₂ ⊚ fAX xA) y → y = Classical.choose ⋯h_left:fAP ((↑f').1 xA) = Classical.choose ⋯h_right:fAP (fA xA) = Classical.choose ⋯⊢ fAP ((↑f').1 xA) = fAP (fA xA)
      rw [h_left, h_right]All goals completed! 🐙
    -- Show dot maps are equal
    ·h.right.hDP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂⊢ (↑f').2 = (↑f).2 funext xDh.right.hD.hP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierD⊢ (↑f').2 xD = (↑f).2 xD
      have hp₁ : p₁ ⊚ fDP (f'.val.2 xD) = f₁ ⊚ fDX xD := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
        apply (homDEquiv B₁).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierD⊢ (homDEquiv B₁) (p₁ ⊚ fDP ((↑f').2 xD)) = (homDEquiv B₁) (f₁ ⊚ fDX xD)
        apply_fun (fun k => k.val.2 xD) at hf'₁aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhf'₁:(↑(p₁ ⊚ f')).2 xD = (↑f₁).2 xD⊢ (homDEquiv B₁) (p₁ ⊚ fDP ((↑f').2 xD)) = (homDEquiv B₁) (f₁ ⊚ fDX xD)
        exact hf'₁All goals completed! 🐙
      have hp₂ : p₂ ⊚ fDP (f'.val.2 xD) = f₂ ⊚ fDX xD := byP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂⊢ ∀ (X : IrreflexiveGraph) (f₁ : X ⟶ B₁) (f₂ : X ⟶ B₂), ∃! f, p₁ ⊚ f = f₁ ∧ p₂ ⊚ f = f₂
        apply (homDEquiv B₂).injectiveaP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xD⊢ (homDEquiv B₂) (p₂ ⊚ fDP ((↑f').2 xD)) = (homDEquiv B₂) (f₂ ⊚ fDX xD)
        apply_fun (fun k => k.val.2 xD) at hf'₂aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xDhf'₂:(↑(p₂ ⊚ f')).2 xD = (↑f₂).2 xD⊢ (homDEquiv B₂) (p₂ ⊚ fDP ((↑f').2 xD)) = (homDEquiv B₂) (f₂ ⊚ fDX xD)
        exact hf'₂All goals completed! 🐙
      apply (homDEquiv P).symm.injectiveh.right.hD.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xDhp₂:p₂ ⊚ fDP ((↑f').2 xD) = f₂ ⊚ fDX xD⊢ (homDEquiv P).symm ((↑f').2 xD) = (homDEquiv P).symm ((↑f).2 xD)
      change fDP (f'.val.2 xD) = fDP (fD xD)h.right.hD.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xDhp₂:p₂ ⊚ fDP ((↑f').2 xD) = f₂ ⊚ fDX xD⊢ fDP ((↑f').2 xD) = fDP (fD xD)
      have h_uniq := (Classical.choose_spec (hD (f₁ ⊚ fDX xD)
          (f₂ ⊚ fDX xD))).2h.right.hD.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xDhp₂:p₂ ⊚ fDP ((↑f').2 xD) = f₂ ⊚ fDX xDh_uniq:∀ (y : D ⟶ P), (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP = f₂ ⊚ fDX xD) y → y = Classical.choose ⋯⊢ fDP ((↑f').2 xD) = fDP (fD xD)
      have h_left := h_uniq (fDP (f'.val.2 xD)) ⟨hp₁, hp₂⟩h.right.hD.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xDhp₂:p₂ ⊚ fDP ((↑f').2 xD) = f₂ ⊚ fDX xDh_uniq:∀ (y : D ⟶ P), (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP = f₂ ⊚ fDX xD) y → y = Classical.choose ⋯h_left:fDP ((↑f').2 xD) = Classical.choose ⋯⊢ fDP ((↑f').2 xD) = fDP (fD xD)
      have h_right := h_uniq (fDP (fD xD)) (hD_proj xD)h.right.hD.h.aP:IrreflexiveGraphB₁:IrreflexiveGraphB₂:IrreflexiveGraphp₁:P ⟶ B₁p₂:P ⟶ B₂hA:∀ (fAP₁ : A ⟶ B₁) (fAP₂ : A ⟶ B₂), ∃! fAP, p₁ ⊚ fAP = fAP₁ ∧ p₂ ⊚ fAP = fAP₂hD:∀ (fDP₁ : D ⟶ B₁) (fDP₂ : D ⟶ B₂), ∃! fDP, p₁ ⊚ fDP = fDP₁ ∧ p₂ ⊚ fDP = fDP₂X:IrreflexiveGraphf₁:X ⟶ B₁f₂:X ⟶ B₂fAX:X.carrierA → (A ⟶ X) := fun xA ↦ (homAEquiv X).symm xAfDX:X.carrierD → (D ⟶ X) := fun xD ↦ (homDEquiv X).symm xDfA:X.carrierA → P.carrierA := 
  fun xA ↦
    let h_uniq := ⋯;
    (homAEquiv P) (Classical.choose h_uniq)fD:X.carrierD → P.carrierD := 
  fun xD ↦
    let h_uniq := ⋯;
    (homDEquiv P) (Classical.choose h_uniq)fAP:P.carrierA → (A ⟶ P) := fun pA ↦ (homAEquiv P).symm pAfDP:P.carrierD → (D ⟶ P) := fun pD ↦ (homDEquiv P).symm pDhA_proj:∀ (xA : X.carrierA), p₁ ⊚ fAP (fA xA) = f₁ ⊚ fAX xA ∧ p₂ ⊚ fAP (fA xA) = f₂ ⊚ fAX xAhD_proj:∀ (xD : X.carrierD), p₁ ⊚ fDP (fD xD) = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP (fD xD) = f₂ ⊚ fDX xDhSrc_comm:∀ (xA : X.carrierA), P.toSrc (fA xA) = fD (X.toSrc xA)hTgt_comm:∀ (xA : X.carrierA), P.toTgt (fA xA) = fD (X.toTgt xA)f:X ⟶ P := ⟨(fA, fD), ⋯⟩f':X ⟶ Phf'₁:p₁ ⊚ f' = f₁hf'₂:p₂ ⊚ f' = f₂xD:X.carrierDhp₁:p₁ ⊚ fDP ((↑f').2 xD) = f₁ ⊚ fDX xDhp₂:p₂ ⊚ fDP ((↑f').2 xD) = f₂ ⊚ fDX xDh_uniq:∀ (y : D ⟶ P), (fun fDP ↦ p₁ ⊚ fDP = f₁ ⊚ fDX xD ∧ p₂ ⊚ fDP = f₂ ⊚ fDX xD) y → y = Classical.choose ⋯h_left:fDP ((↑f').2 xD) = Classical.choose ⋯h_right:fDP (fD xD) = Classical.choose ⋯⊢ fDP ((↑f').2 xD) = fDP (fD xD)
      rw [h_left, h_right]All goals completed! 🐙

Exercise 2 (p. 252)

What is a figure of shape

abbrev A₂ : IrreflexiveGraph := {
  carrierA := Fin 2
  carrierD := Fin 3
  toSrc := fun
    | 0 => 0
    | 1 => 1
  toTgt := fun
    | 0 => 1
    | 1 => 2
}

in a graph X? What are the various ways in which it can be singular?

Solution: Exercise 2

A figure of shape A_2 in a graph X is a morphism {A_2 \xrightarrow{f} X} that maps the two arrows of A_2 to two arrows (not necessarily distinct) in X and the three dots of A_2 to three dots (not necessarily distinct) in X in such a way as to preserve sources and targets.

def FigA₂ (X : IrreflexiveGraph) := A₂ ⟶ X

The figure is singular if the two arrows of A_2 are mapped to the same arrow in X or if any two of the three dots of A_2 are mapped to the same dot in X—that is, if the morphism f fails to be injective on either the arrows or the dots.

def FigA₂.IsSingular {X : IrreflexiveGraph} (f : FigA₂ X) : Prop :=
  ¬(Function.Injective f.val.1 ∧ Function.Injective f.val.2)

Specifically, the figure is singular if any of the following conditions hold:

(1) the two arrows of A_2 are mapped to the same arrow in X

example {X : IrreflexiveGraph} (f : FigA₂ X)
    (h : f.val.1 0 = f.val.1 1) : f.IsSingular := byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).1 0 = (↑f).1 1⊢ f.IsSingular
  intro ⟨hA, _⟩X:IrreflexiveGraphf:FigA₂ Xh:(↑f).1 0 = (↑f).1 1hA:Function.Injective (↑f).1right✝:Function.Injective (↑f).2⊢ False
  exact absurd (hA h) (byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).1 0 = (↑f).1 1hA:Function.Injective (↑f).1right✝:Function.Injective (↑f).2⊢ ¬0 = 1 decideAll goals completed! 🐙)

(2) the first and second dots of A_2 are mapped to the same dot in X

example {X : IrreflexiveGraph} (f : FigA₂ X)
    (h : f.val.2 0 = f.val.2 1) : f.IsSingular := byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 0 = (↑f).2 1⊢ f.IsSingular
  intro ⟨_, hD⟩X:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 0 = (↑f).2 1left✝:Function.Injective (↑f).1hD:Function.Injective (↑f).2⊢ False
  exact absurd (hD h) (byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 0 = (↑f).2 1left✝:Function.Injective (↑f).1hD:Function.Injective (↑f).2⊢ ¬0 = 1 decideAll goals completed! 🐙)

(3) the first and third dots of A_2 are mapped to the same dot in X

example {X : IrreflexiveGraph} (f : FigA₂ X)
    (h : f.val.2 0 = f.val.2 2) : f.IsSingular := byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 0 = (↑f).2 2⊢ f.IsSingular
  intro ⟨_, hD⟩X:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 0 = (↑f).2 2left✝:Function.Injective (↑f).1hD:Function.Injective (↑f).2⊢ False
  exact absurd (hD h) (byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 0 = (↑f).2 2left✝:Function.Injective (↑f).1hD:Function.Injective (↑f).2⊢ ¬0 = 2 decideAll goals completed! 🐙)

(4) the second and third dots of A_2 are mapped to the same dot in X

example {X : IrreflexiveGraph} (f : FigA₂ X)
    (h : f.val.2 1 = f.val.2 2) : f.IsSingular := byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 1 = (↑f).2 2⊢ f.IsSingular
  intro ⟨_, hD⟩X:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 1 = (↑f).2 2left✝:Function.Injective (↑f).1hD:Function.Injective (↑f).2⊢ False
  exact absurd (hD h) (byX:IrreflexiveGraphf:FigA₂ Xh:(↑f).2 1 = (↑f).2 2left✝:Function.Injective (↑f).1hD:Function.Injective (↑f).2⊢ ¬1 = 2 decideAll goals completed! 🐙)

The case in which all three dots of A_2 are mapped to the same dot in X is covered by any of conditions (2), (3), or (4), since if all three dots are mapped to the same dot, then in particular any two dots are mapped to the same dot.