Is the morphism $\mathcal F^\dagger \to \mathcal G^\dagger$ induced by $f$ injective?

Question

Let $f: \mathcal F\to \mathcal G$ be an injective morphism of presheaves, is the morphism $\mathcal F^\dagger \to \mathcal G^\dagger$ induced by $f$ injective?

Answer 1

Yes. If $f$ is injective between $\mathcal{F}$ and $\mathcal{G}$, then the stalks of $f$ are injective. Therefore, the stalks of $f^{\dagger}$ are injective. But on sheafs, this is sufficient to prove the morphism injective.