A question in "Conceptual Mathematics" In Conceptual Mathematics 1st edition, p. 325-236, there is a sketch of a proof, but I can't carry out the complete proof.

"... This also follows from the appropriate universal mapping properties, which imply that the two composites satisfy properties that only the corresponding identity maps satisfy."

I can't figure this out.
Can you give me a clue?

A product $A\times B$ is a triple $(A,B,A\times B)$ equipped with maps $\pi_A\colon A\times B\to A$, and $\pi_B\colon A\times B\to B$ such that given any pair of maps $\psi_A\colon X\to A$ and $\psi_B\colon X\to B$, there is a unique map $\psi\colon X\to A\times B$, such that $\psi_A=\pi_a\circ \psi$ and $\psi_B=\pi_B\circ \psi$. Hence for $X = A\times B$, (and some given maps) if the identity map $A\times B\to A\times B$ fulfills this role, then any other map also fulfilling it equals the identity map.
I.e. to prove that $fg$ is the identity, check that it does satisfy some property that is only satisfied by one unique map, and then check that the identity also satisfies it.  Examples are given in the algebra notes 843-4-5 on my website.
Specifically, look on pages 8-9 of the notes 845-3, for exactly this proof of uniqueness of products.
