Lemma 99.21.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks. Then

For every DM algebraic stack $\mathcal{Z}$ and morphism $\mathcal{Z} \to \mathcal{Y}$ there exists a scheme and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.

For every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ there exists a scheme and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} Z$.

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