 Stalk functor for presheaves of sets

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.