Skyscraper sheaf of sets ($i_{\ast }S(V)=S\text{ if }x\in V,i_{\ast }S(V)=1\text{ otherwise}$)

For inclusion of a point $i:\{x\}\to X$, this is the direct-image sheaf of a set. It is right adjoint to the stalk functor. It uses the singleton terminal set on opens omitting $x$, rather than the zero group used for abelian-group skyscraper sheaves.