The Stacks project

Lemma 97.4.1. Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ are representable by algebraic spaces and étale so is $\mathcal{X} \to \mathcal{Y}$.

Proof. Let $\mathcal{U}$ be a representable category fibred in groupoids over $S$. Let $f : \mathcal{U} \to \mathcal{Y}$ be a $1$-morphism. We have to show that $\mathcal{X} \times _\mathcal {Y} \mathcal{U}$ is representable by an algebraic space and étale over $\mathcal{U}$. Consider the composition $h : \mathcal{U} \to \mathcal{Z}$. Then

\[ \mathcal{X} \times _\mathcal {Z} \mathcal{U} \longrightarrow \mathcal{Y} \times _\mathcal {Z} \mathcal{U} \]

is a $1$-morphism between categories fibres in groupoids which are both representable by algebraic spaces and both étale over $\mathcal{U}$. Hence by Properties of Spaces, Lemma 66.16.6 this is represented by an étale morphism of algebraic spaces. Finally, we obtain the result we want as the morphism $f$ induces a morphism $\mathcal{U} \to \mathcal{Y} \times _\mathcal {Z} \mathcal{U}$ and we have

\[ \mathcal{X} \times _\mathcal {Y} \mathcal{U} = (\mathcal{X} \times _\mathcal {Z} \mathcal{U}) \times _{(\mathcal{Y} \times _\mathcal {Z} \mathcal{U})} \mathcal{U}. \]

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