The Stacks project

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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