I am trying to understand the hom-tensor relation and as usual I'm having a hard time defining functors on the functions.
So I have a functor $$(F:= ){_A\text{Hom}}(-,-):{_A\text{Mod}_B}^{op}\times {_A\text{Mod}_C}\rightarrow {_B\text{Mod}_C}$$ the actions on objects is clear: $$F(M,N) = {_A\text{Hom}(M,N)}$$ However, what is the action on functions? I take $$f\in \text{Hom}_{{_A\text{Mod}_B}^{op}}(M,M'),g\in \text{Hom}_{{_A\text{Mod}_C}}(N,N')$$ Then $(f,g)\in \text{Hom}_{{{_A\text{Mod}_B}^{op}}\times{_A\text{Mod}_C}}((M,N),(M',N'))$
and want to define $$F(f,g):{_A\text{Hom}(M,N)}\rightarrow {_A\text{Hom}(M',N')}$$ How does this work?
EDIT
after writing things down properly, it makes sense:
if $h$ is from $M$ to $N$ then define it's image as $g\circ h\circ f$