Lemma 101.33.8. Let \pi : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. If \mathcal{X} is a gerbe over \mathcal{Y}, then \pi is surjective and smooth.
Proof. We have seen surjectivity in Lemma 101.28.8. By Lemma 101.33.4 it suffices to prove to the lemma after replacing \pi by a base change with a surjective, flat, locally finitely presented morphism \mathcal{Y}' \to \mathcal{Y}. By Lemma 101.28.7 we may assume \mathcal{Y} = U is an algebraic space and \mathcal{X} = [U/G] over U with G \to U flat and locally of finite presentation. Then we win by Lemma 101.33.7. \square
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