Lemma 100.43.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

1. $f$ is proper, and

2. $f$ satisfies both the uniqueness and existence parts of the valuative criterion.

Proof. A proper morphism is the same thing as a separated, finite type, and universally closed morphism. Thus this lemma follows from Lemmas 100.41.2, 100.41.3, 100.42.1, and 100.42.2. $\square$

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