The Stacks project

Lemma 101.28.5. Let

\[ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

be a fibre product of algebraic stacks. If $\mathcal{Y}' \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation and $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$, then $\mathcal{X}$ is a gerbe over $\mathcal{Y}$.

Proof. Follows immediately from Lemma 101.27.13 and Stacks, Lemma 8.11.7. $\square$

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