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