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.

For the second question: I believe that "natural" here means canonical, since it comes out of the universal property of sheafification. I realise that this is confusing since "natural" has a specific meaning regarding compatibility with functors in category theory.

  • $\begingroup$ I thought so, but it is weird, because canonical and natural aren't really equivalent qualifications and I could understand where the canonical would come from in this case, but not the natural. Thanks $\endgroup$ – Dalamar Oct 26 '18 at 14:46

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