Lemma 100.5.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Y}' \to \mathcal{Y}$ be a surjective morphism of algebraic stacks. If the base change $f' : \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective.

**Proof.**
Immediate from Lemma 100.4.3.
$\square$

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