Lemma 101.28.4 (06R8)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.28: Gerbes
Lemma 101.28.4 (cite)

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Lemma 101.28.4. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{Y}$ is a gerbe over $\mathcal{Z}$ , then $\mathcal{X}$ is a gerbe over $\mathcal{Z}$ .

Proof. Immediate from Stacks, Lemma 8.11.6. $\square$

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