Is a morphism of sheaves of abelian groups epi iff it is epi as a morphism of sheaves of sets?

Question

Let $T=Sh^{Sets}(C)$ be the category of sheaves on a site $C$ and $S=Sh^{Ab}(C)$ be the category of abelian group objects in $T$, this is sheaves on $C$ with values in abelian groups.

Is a morphism $f:X\to Y$ in $S$ an epimorphism if and only if it is an epimorphism as a morphism in $T$ (after applying the forgetful functor)?

Answer 1

Yes. In both cases, $f$ is an epi iff for every element of $Y(x)$ there is a covering $\{x_i \to x\}$ such that the restrictions to $Y(x_i)$ have a preimage in $X(x_i)$. Basically, the reason is that the forgetful functor preserves images.