What is an example of a Grothendieck topology $J$ on a small category $\mathcal{C}$ such that the category of sheaves $\mathsf{Sh}(\mathcal{C},J)$ has a monomorphism which is not a monomorphism of presheaves (i.e. is not injective on sections)? Clearly $J$ cannot be subcanonical.
Background. In the book by Mac Lane - Moerdijk the proof that $\mathsf{Sh}(\mathcal{C},J)$ is an elementary topos, specifically in the proof that $\Omega_J$ is a subobject classifier, it seems to be used that subsheaves coincide with subobjects, without any assumption on $J$.
The inclusion $i \colon \mathsf{Sh}(\mathcal C,J) \hookrightarrow \hat{\mathcal C}$ has a left adjoint, namely the sheafification functor, hence preserves monomorphisms.