Lemma 94.9.6 (04SY)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 94: Algebraic Stacks
Section 94.9: Morphisms representable by algebraic spaces
Lemma 94.9.6 (cite)

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Lemma 94.9.6. Let $S$ be an object of $\mathit{Sch}_{fppf}$ . Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$ -morphism of categories fibred in setoids over $(\mathit{Sch}/S)_{fppf}$ . Let $F$ , resp. $G$ be the presheaf which to $T$ associates the set of isomorphism classes of objects of $\mathcal{X}_ T$ , resp. $\mathcal{Y}_ T$ . Let $a : F \to G$ be the map of presheaves corresponding to $f$ . Then $a$ is representable by algebraic spaces (see Bootstrap, Definition 80.3.1) if and only if $f$ is representable by algebraic spaces.

Proof. Omitted. Hint: Combine Lemmas 94.9.3 and 94.9.5. $\square$

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