Let $C,D$ be categories, where $C$ is small. If we have morphism $\eta$ (i.e. natural transformation) in the category of functors $D^C$, then clearly $\eta$ is a monomorphism if all components $\eta_x$, $x \in C$ are monomorphisms. The converse is true if $D$ has pullbacks:
It is well-known that $D^C$ also has pullbacks, computed pointwise. Let $\eta : F \to G$ and consider the diagonal $\Delta : F \to F \times_G F$. Then: $\eta$ is a monomorphism iff $\Delta$ is an isomorphism iff all $\Delta_x : F(x) \to F(x) \times_{G(x)} F(x)$, $x \in C$ are isomorphisms iff all $\eta_x : F(x) \to G(x)$ are monomorphisms.
Question. Can you give an example where the converse fails, i.e. a monomorphism in the category of functors which is not pointwise a monomorphism?
This is a related question.