The set products satisfy the coproduct property in the Abelian category.

The set product, where elements are pairs, is capable of acting as a co-product object in the category $\mathrm{Ab}$. In other words, the following diagram commutes in $\mathrm{Ab}$ where $\sigma$ is uniquely determined by the rest of the diagram.

$i_{(G,e_H)}$ and $i_{(e_G,H)}$, the inclusion functions

The function $i_{(G,e_H)}:G\to G\times H$ and $i_{(e_G,H)}:H\to G\times H$ are special inclusion functions given by:

They place the input in either the first or second entry of a pair, leaving the other entry to be the identity element of the corresponding group, $G$ or $H$. The fixed identity entry allows the variable entry to inherit the group behaviour of the input. This insures that the inclusion functions satisfy the requirements to be [what?].

The morphism $\sigma$

It is not a matter of searching for $\sigma$, for it's form was anticipated from the beginning.

$\sigma :G\times H\to Z$ can be expressed precisely in terms of $f_1,f_2$ and ${\cdot }_Z$:

The definition of $\sigma$ follows (is forced from) from the constraints. What is interesting is that $\sigma$ is actually possible! And only just. Can you remember why $\sigma$ works in $\mathrm{Ab}$ but not in $\mathrm{Grp}$?