Joint stalk functor on sheaves of sets
= Joint stalk functor on sheaves of sets
{title2=$F\mapsto(F_x)_{x\in X}$}
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.