Let $C$ and $D$ be categories and let $F : C \rightarrow D$, $G : C \rightarrow D$ be two functors such that they are either both covariant or both contravariant. Under what most general hypotheses is the following statement always true:
A natural transformation $\eta : F \rightarrow G$ is a monomorphism (epimorphism) iff each component of the the natural transformation is a monomorphism (epimorphism).
What I would like to know are the weakest possible restrictions on $C$ and $D$ for which the forward implication holds.
Also, can someone direct me to a source that discusses this?
I should perhaps mention how I came about this question. I was trying to show that if $X$ is a topological space and $F, G$ are sheaves of sets on $X$, then a morphism of sheaves $\eta: F \rightarrow G$ is a monomorphism if and only if the induced maps on stalks are injective (**).
There is probably a way to prove this without using the fact that the components of $\eta$ are monic (i.e. injective set functions in this case), but I already know from an argument using the Yoneda Lemma that in this particular case because my target category is the category of sets, $\eta$ is a monomorphism if and only if each of its components are (and the proof of (**) is quite elementary, once I know this fact).
Also, I am not aware of a proof of the fact that $\eta$ is an epimorphism iff every component of $\eta$ is an epimorphism, even when the target category is the category of sets. So, is this statement true when the target category is the category of sets?
(Later edit: After starting a bounty on this question, I realized that it might not be possible to give the weakest possible conditions on the categories $C$ and $D$, like I am looking for. In that case, any known conditions on $C$ and $D$ for which my question holds will also suffice.)