Sheaf on a site

A categorical presheaf is a sheaf for $J$ if every matching family on a covering sieve has a unique amalgamation. Equivalently restriction $\mathrm{Nat}(yU,F)\to \mathrm{Nat}(S,F)$ is bijective for each covering sieve $S\hookrightarrow yU$. Sheaves on a small site form a Grothendieck topos.

A contravariant set-valued functor on schemes is a Zariski sheaf if compatible local elements on any Zariski cover glue uniquely. Explicitly, $F(T)$ is the equalizer of ${\prod }_{i}F(U_i)\rightrightarrows {\prod }_{i,j}F(U_i{\times }_T{U}_j)$. The functor of points of every scheme is a Zariski sheaf because morphisms of schemes glue over open covers.