I've been trying to work through the following problem:
Let $H$, $G$, and $G'$ be groups, and let $f:H\to G$ and $g:H\to G'$ be two homomorphisms. Define the notion of coproduct of these two homomorphisms over $H$, and show that it exists.
I'm new to category theory and free groups, so I'm pretty lost on this. Here's what I do know:
Definition. If $\{A_{j}\}_{j\in J}$ is a family of objects in a category $\mathscr{A}$, then their coproduct is a pair $(S,\{f_{j}\}_{j\in J})$ consisting of an object $S$ and a family of morphisms $\{f_{j}:A_{j}\to S\}$ such that, given a family of morphisms $\{g_{j}:A_{j}\to T\}$, there exists a unique morphism $h:S\to T$ such that $h\circ f_{j}=g_{j}$ for all $j\in J$.
I'm really confused as to how to apply this definition. I'm assuming that I need to take my category to be the category of all groups, but I'm not sure. Any help is greatly appreciated!
Look at the free product of $G$ and $G^\prime$ where you identify $f(h)$ with $g(h)$.