Hom of sheafification

Question

Let $\mathcal{F},\mathcal{G}$ be presheaf of modules. Can we say that there exist a bijection between $\hom(\mathcal{F},\mathcal{G})$ and $\hom_{\mathrm{sheaf}}(\mathcal{F}^{\sharp},\mathcal{G}^{\sharp})$.

Answer 1

No. As a simple example, consider the empty space $X = \emptyset$. Then every module defines a presheaf on $X$, but only the zero module defines a sheaf on $X$. In particular every sheafification gives the $0$ sheaf. Therefore we also only have trivial morphisms of sheaves on the level of sheafifications, but may have non-trivial morphisms before passing to the sheafification.

The "right" bijection one has is the one coming from the sheafification being the left adjoint of the forgetful functor. So there is always a natural bijection

$$\text{Hom}_{\text{Sh}}(\mathcal{F}^{\sharp},\mathcal{G}) = \text{Hom}_{\text{PSh}} (\mathcal{F},\mathcal{G})$$ for every presheaf $\mathcal{F}$ and every sheaf $\mathcal{G}$. In particular, we have a natural bijection $$\text{Hom}_{\text{Sh}}(\mathcal{F}^{\sharp},\mathcal{G}^{\sharp}) = \text{Hom}_{\text{PSh}} (\mathcal{F},\mathcal{G}^{\sharp})$$ if $\mathcal{G}$ is a presheaf.