Stalk functor for presheaves of sets ($F\mapsto F_x$)

For presheaves of sets, $F_x={\underrightarrow{\lim }}_{U\ni x}F(U)$. This functor preserves finite limits because the neighborhood system is filtered and filtered colimits commute with finite limits in sets. Its right adjoint sends a set $S$ to the presheaf taking the value $S$ on opens containing $x$ and the singleton on other opens.

The joint stalk functor $\mathbf{Sh}(X)\to {\mathbf{Set}}^X$ is faithful: equality of germs makes two images of a section locally equal, and the sheaf uniqueness axiom makes them globally equal. Its right adjoint is the product of the set-valued skyscraper sheaves at all points. The Beck comonadicity theorem then makes the joint stalk functor comonadic.