Exercise from Algebra of Programming on catamorphisms

Question

This is an exercise from Algebra of Programming which I'm reading for self-study. Below, $T$ is the initial algebra of $F$.

What I've tried:

I can construct $h \circ F (\pi_2) : F(A \times B) \rightarrow B$. Then we take the product $\langle g, h \circ F(\pi_2) \rangle : F(A \times B) \rightarrow A \times B$. We can then take the catamorphism $cata(\langle g, h \circ F(\pi_2) \rangle) : T \rightarrow A\times B$ and need to show that $f = \pi_1 \circ cata(\langle g, h \circ F(\pi_2) \rangle)$.

I've tried using the fact that the diagram above commutes to get the equality above, to no avail. Any suggestions?

Answer 1

The solution as per @Colin McQuillan's hint above:

We show that the product $\langle f, cata(h) \rangle$ is equal to the catamorphism $cata(\langle g, h \circ F(outr) \rangle)$.

By the universal property of catamorphisms, it suffices to show that

$\langle f, cata(h) \rangle \circ \alpha = \langle g, h \circ F(outr) \rangle \circ F \langle f, cata(h) \rangle$

On the left, we have

$\langle f, cata(h) \rangle \circ \alpha = \langle f \circ \alpha, cata(h) \circ \alpha \rangle = \langle g \circ F \langle f, cata(h) , h \circ F (cata(h)) \rangle \rangle$

Notice that $F(cata(h)) = F outr \circ F \langle f, cata(h) \rangle$, so

$\langle g \circ F \langle f, cata(h) , h \circ F (cata(h)) \rangle \rangle = \langle g \circ F \langle f, cata(h) \rangle, h \circ F outr \circ F \langle f, cata(h) \rangle\rangle = \langle g, h \circ F outr \rangle \circ F \langle f, cata(h) \rangle$

as needed. $\blacksquare$