The Stacks project

Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{smooth} \to (\mathit{Sch}/S)_{smooth}$. Being cocontinuous just means that any smooth covering of $T/S$, $T$ affine, can be refined by a standard smooth covering of $T$. This is the content of Lemma 34.5.4. Hence (1) holds. We see $u$ is continuous simply because a standard smooth covering is a smooth covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering. $\square$