Let $\eta:F\rightarrow G$ be a morphism of sheaves. In the notes of a class of mine it is emphasized that while $\eta$ is monic iff all its components are monic, there is no corresponding equivalence for epis, and we must resort to the germs.
But the dual of the general fact below seems to imply otherwise since $\mathsf{Set}$ is bicomplete. What am I missing here?
It's true that whether a morphism of presheaves is an epimorphism can be checked componentwise, but the inclusion functor from sheaves to presheaves doesn't preserve epimorphisms. It's a right adjoint, so it preserves limits, which implies that it preserves monomorphisms, but that's all you get for free.