Let $D:A \rightarrow B$ be a morphism in Category $C$. Define two functors $C \rightarrow Sets$ by the rules $F(X) = mor_C (X,A)$ and $G(X) = mor_C(X,B)$. Show that $H:F \rightarrow G$, where $H_X:F(X) \rightarrow G(X)$ is defined as $H_X(f) = D o f$, is a natural transformation.
I have attempted to prove this by directly writing out the natural transformation definition and just showing that the diagram is commutative but the problem I am currently facing is that the two covariant functors, $F$ and $G$, are not defined explicitly at the specific morphisms and so I cannot further simplify my diagram chasing. It would be great if someone could provide some hints.Thanks.