Sheafification as a Kan Extension of the Identity?

Question

How can the sheafification functor be described in terms of a Kan extension of the identity on the category of $\mathsf{Set}$-valued sheaves (over some topological space)? How about general $\mathsf C$-valued sheaves?

Answer 1

Sheafification is left adjoint to the inclusion functor from sheaves to presheaves.

If $F : C \to D$ is left adjoint to $G : D \to C$ with unit $\eta$ and counit $\varepsilon$, then $F = \mathrm{Ran}_G(\mathrm{id}_D)$, because for $H : C \to D$ we have $$\hom(H \circ G , \mathrm{id}_D) \cong \hom(H,F).$$ More generally, we have $$\hom(H \circ G,G') \cong \hom(H,G' \circ F)$$ for functors $H : C \to D'$, $G' : D \to D'$.

In fact, $H \circ G \to G'$ is mapped to $H = H \circ \mathrm{id}_C \xrightarrow{H \bullet \eta} H \circ G \circ F \to G' \circ F$, and conversely, $H \to G' \circ F$ is mapped to $H \circ G \to G' \circ F \circ G \xrightarrow{G' \bullet \varepsilon} G' \circ \mathrm{id}_D = G'$.