From Mac Lane's Category Theory:
I'm trying to prove the below exercise but I'm having trouble. I can see that (assuming the common left inverse exists) small products implies using Theorem $1$ (Included in the picture below) and the result follows.
But I can't see what the common left inverse is.
Anyone have any ideas?
The common left inverse comes from the identities of $J$.
More precisely : for every object $i$ of $J$ one has an identity which is an arrow $1_i:i\to i$ in $J$, and for each of these identity arrows one has a corresponding projection $\pi_{1_i}:\prod_{u}F_{\operatorname{cod} u}\to F_{\operatorname{cod}1_i}=F_i$. Let $h:\prod_{u}F_{\operatorname{cod} u}\to \prod_i F_i$ be the arrow given by the universal property of the product $\prod_i F_i$. Then in order to check that $h$ is a common left inverse to $f$ and $g$ it is enough to check that $\pi_ihf=\pi_i=\pi_ihg$ for all $i$. Now by definition one has $$\pi_ihf=\pi_{1_i}f=\pi_{\operatorname{cod}1_i}=\pi_i$$ and $$\pi_ihg=\pi_{1_i}g=F(1_i)\pi_{\operatorname{dom}1_i}=\pi_i,$$ which concludes the proof.