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 \( \cat{Ab} \). In other words, the following diagram commutes in \( \cat{Ab} \) where \( \sigma \) is uniquely determined by the rest of the diagram.
[:can you remember the commutative diagram?]
\( i_{\small{(G, e_H)}} \) and \( i_{\small{(e_G, H)}} \), 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 \( \groupMul{Z} \):
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 \( \cat{Ab} \) but not in \( \cat{Grp} \)?