We proceed by repeatedly applying the property {f \circ \alpha = \beta \circ f}.

The automaton modelling the behaviour of a light bulb that can only be lit four times before burning out may be represented as follows:

To verify correct operation, we begin by setting up a simulation of pressing the on-off button eight times, starting from off₀, and counting how many times the bulb lights up.

Run the simulation starting with a lit count of 0, and return the final bulb state and lit count.

Confirm that at the end of the run, the bulb has been lit four times and is now burntOut.

Lastly, confirm that once the bulb is burntOut, pressing the on—off button has no effect (i.e., burntOut is a fixed point).

(a) For \mathbf{P} in 𝑺↻↻, we first define a Person type to use in place of the set P of all people, and we identity each Person as being either a mother or a father.

We next define the mother and father endomaps on Person, which ignore their input (as irrelevant) and simply return a Person of the appropriate ParentType.

Now we can define the object \mathbf{P}.

Lastly, we define the map gender, which sends each Person to their corresponding Gender.

Since we can form a valid morphism using our categorical framework, it follows that gender is a map in the category 𝑺↻↻ from \mathbf{P} to \mathbf{G}.

For good measure, we can also verify the two commutative properties explicitly.

(b) The implementation of part (b) is similar to that of part (a), with an appropriate change to our Person structure to handle clan.

(c) The implementation of part (c) is a combination of parts (a) and (b).

As required, we define appropriate 'mother' and 'father' maps (m and f, respectively, to align with the book).