The Stacks project

Proof. We will prove this by verifying (1) and (2') of Topologies, Lemma 34.10.12. The sheaf property for Zariski coverings follows from the fact that $F$ has the sheaf property for all h coverings. Finally, suppose that $X \to Y$ is a morphism of affine schemes over $S$ such that $\{ X \to Y\} $ is a V covering. By Lemma 61.10.2 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\} $ is an h covering for each $i$. We obtain

which is what we wanted to show. The first equality because $F$ is limit preserving and $X = \mathop{\mathrm{lim}}\nolimits X_ i$ and $X \times _ Y X = \mathop{\mathrm{lim}}\nolimits X_ i \times _ Y X_ i$. The second equality because filtered colimits are exact. The third equality because $F$ satisfies the sheaf property for h coverings. $\square$