Lemma 94.9.4 (02ZZ)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 94: Algebraic Stacks
Section 94.9: Morphisms representable by algebraic spaces
Lemma 94.9.4 (cite)

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Lemma 94.9.4. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $S$. If $\mathcal{X}$ and $\mathcal{Y}$ are representable by algebraic spaces over $S$, then the $1$-morphism $f$ is representable by algebraic spaces.

Proof. Omitted. This relies only on the fact that the category of algebraic spaces over $S$ has fibre products, see Spaces, Lemma 65.7.3. $\square$

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