Lemma 69.20.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated and of finite type. The following are equivalent

The morphism $f$ is proper.

For any morphism $Y \to Z$ which is locally of finite presentation the map $|X \times _ Y Z| \to |Z|$ is closed, and

there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $|\mathbf{A}^ n \times (X \times _ Y V)| \to |\mathbf{A}^ n \times V|$ is closed for all $n \geq 0$.

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