The Stacks project

Lemma 101.28.3. 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 $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$.

Proof. Immediate from the definitions and Stacks, Lemma 8.11.5. $\square$

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