The Stacks project

Lemma 97.4.2. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Suppose that $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Z} \to \mathcal{Y}$ are $1$-morphisms. If

  1. $\mathcal{Y}$, $\mathcal{Z}$ are representable by algebraic spaces $Y$, $Z$ over $S$,

  2. the associated morphism of algebraic spaces $Y \to Z$ is surjective, flat and locally of finite presentation, and

  3. $\mathcal{Y} \times _\mathcal {Z} \mathcal{X}$ is a stack in setoids,

then $\mathcal{X}$ is a stack in setoids.

Proof. This is a special case of Stacks, Lemma 8.6.10. $\square$

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