Lemma 64.11.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $G$ be an algebraic space over $S$, let $F$ be a sheaf on $(\mathit{Sch}/S)_{fppf}$, and let $G \to F$ be a representable transformation of functors which is surjective and étale. Then $F$ is an algebraic space.

Proof. Pick a scheme $U$ and a surjective étale morphism $U \to G$. Since $G$ is an algebraic space $U \to G$ is representable. Hence the composition $U \to G \to F$ is representable, surjective, and étale. See Lemmas 64.3.2 and 64.5.4. Thus $F$ is an algebraic space by Lemma 64.11.1. $\square$

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