Lemma 99.23.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

$f$ is quasi-DM,

for any morphism $V \to \mathcal{Y}$ with $V$ an algebraic space there exists a surjective, flat, locally finitely presented, locally quasi-finite morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ where $U$ is an algebraic space, and

there exist algebraic spaces $U$, $V$ and a morphism $V \to \mathcal{Y}$ which is surjective, flat, and locally of finite presentation, and a morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite.

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