Lemma 94.9.9 (0301)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 94: Algebraic Stacks
Section 94.9: Morphisms representable by algebraic spaces
Lemma 94.9.9 (cite)

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Lemma 94.9.9. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ are $1$-morphisms representable by algebraic spaces, then

\[ g \circ f : \mathcal{X} \longrightarrow \mathcal{Z} \]

is a $1$-morphism representable by algebraic spaces.

Proof. This follows from Lemma 94.9.8. Details omitted. $\square$

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