The Stacks project

Lemma 38.35.4. Let $S$ be a scheme contained in a big site $\mathit{Sch}_ h$. Let $F : (\mathit{Sch}/S)_ h^{opp} \to \textit{Sets}$ be an h sheaf satisfying property (b) of Topologies, Lemma 34.13.1 with $\mathcal{C} = (\mathit{Sch}/S)_ h$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all h coverings and is limit preserving (Limits, Remark 32.6.2).

Proof. This is proven by the arguments given in the proofs of Topologies, Lemmas 34.13.3 and 34.13.4 using Lemmas 38.35.3 and 38.34.12. Details omitted. $\square$

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