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 38.34.11. Details omitted.
$\square$

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