Does every morphism $\mathcal F\to \mathcal G$ induce a morphism of sheaves $\mathcal F^{\dagger}\to \mathcal G^{\dagger}$?

Question

Let $\mathcal F,\mathcal G$ be presheaves of Abelian groups on topological space $X$. Does every morphism $\mathcal F\to \mathcal G$ induce a morphism of sheaves $\mathcal F^{\dagger}\to \mathcal G^{\dagger}$?

Answer 1

Yes, sheafification is a functor. It is left adjoint to the forgetful functor from sheaves to presheaves.