Lemma 100.48.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{X}$ is decent, then $\mathcal{Y}$ is decent.

**Proof.**
Assume $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{X}$ is decent. Note that $f$ is a universal homeomorphism by Lemma 100.28.13. Thus the lemma follows from Lemma 100.48.4.
$\square$

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