Let $\mathcal C$ be a category. Recall that a morphism $f : X \to Y$ is **epi** if $$\circ f: \hom(Y,Z) \to \hom(X,Z)$$ is injective for each object $Z \in \mathcal C$. ($f$ is **mono** if $f\circ : \hom(Z,X) \to \hom(Z,Y)$ is injective.)

Let $\mathcal C,\mathcal D$ be categories. Then $\hom(\mathcal C,\mathcal D)$, the collectional of all functors $\mathcal C \to \mathcal D$, is naturally a category, where the morphisms are **natural transformations**: if $F,G: \mathcal C \to \mathcal D$ are functors, a natural transformation $\alpha: F \Rightarrow G$ assigns a morphism $\alpha(x) : F(x) \to G(x)$ in $\mathcal D$ for each object $x \in \mathcal C$, and if $f: x \to y$ is a morphism in $\mathcal C$, then $\alpha(y) \circ F(f) = G(f) \circ \alpha(x)$ as morphisms in $\mathcal D$.

Given a natural transformation, can I check whether it is epi (or mono) by checking pointwise? I.e.: is a natural transformation $\alpha$ epi (mono) iff $\alpha(x)$ is epi (mono) for each $x$?

If not, is there an implication in one direction between whether a natural transformation is epi and whether it is pointwise-epi?

A more general question, one that I never really learned, is what types of properties of a functor are "pointwise" in that they hold for the functor if they hold for the functor evaluated at each object. E.g.: is the (co)product of functors the pointwise (co)product?