Let $X$ be a small site and $C$ a bicomplete (i.e. complete and cocomplete) category (or rather assume enough so that associated sheaves exists; see the comments). Consider the functor $\text{PSh}(X;C) \to \text{Sh}(X;C), F \mapsto F'$, which takes a presheaf to its associated sheaf. To what extent it is natural in $C$? I.e. for which funcor $\alpha : C \to D$ is the canonical map $\alpha(F)' \to \alpha(F')$ an isomorphism for all presheaves $F$ on $X$?

It seems to me that the explicit construction of the associated sheaf shows that this is true if $\alpha$ is continuous and preserves directed colimits, right? Is there a proof which just uses universal properties? I imagine a proof which uses adjoint functors to $\alpha$ in extended categories.

Do you know of any example of a functor where the above map is not an isomorphism?

Similary we may ask these questions about morphisms in $X$. For example if $i : U \to X$ is an open immerson of topological spaces, then $i^*$ commutes with taking associated sheaves. What are other examples of such morphisms, or is there even a characterization?