# Morphism of sheaves property

Two quick questions:

1. Let $$f$$ be a morphism between sheaves $$\mathcal{F}$$ and $$\mathcal{G}$$. Then we know that $$\textrm{im}(f)$$ is in general a presheaf. I believe the inclusion $$i:\textrm{im}(f)(U) \rightarrow G(U)$$ an injective morphism of presheaves for all $$U$$. Is that correct?
2. In a setting like above, I proved that there is an isomorphism between the sheafification of the image of $$f$$ presheaf and a sub sheaf of $$\mathcal{G}$$. However, I saw some call it a "natural" isomorphism. Why is it natural?

For the first question: yes, the morphism of presheaves $$i$$ is injective, as its kernel is zero.