I am working on a homework problem for a class and found that the part I am stuck on can be reduced to the following: Let $u: M \to M'$ and $v: N \to N'$ be module homomorphisms. I am trying to understand the map $(u,v): \text{Hom}(M, N) \to \text{Hom}(M', N')$. My main question is: if $f \in \text{Hom}(M,N)$, how do I determine what $(u,v)(f)$ is?
My initial guess was that its a function that results in a commutative square but wasn't able to show that the function I had in mind was well-defined.