Stalk functor for presheaves of sets
= Stalk functor for presheaves of sets
{title2=$F\mapsto F_x$}
For presheaves of sets, $F_x=\varinjlim_{U\ni x}F(U)$. 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.