The Stacks project

Lemma 29.41.4. The composition of proper morphisms is proper. The same is true for universally closed morphisms.

Proof. A composition of closed morphisms is closed. If $X \to Y \to Z$ are universally closed morphisms and $Z' \to Z$ is any morphism, then we see that $Z' \times _ Z X = (Z' \times _ Z Y) \times _ Y X \to Z' \times _ Z Y$ is closed and $Z' \times _ Z Y \to Z'$ is closed. Hence the result for universally closed morphisms. We have seen that “separated” and “finite type” are preserved under compositions (Schemes, Lemma 26.21.12 and Lemma 29.15.3). Hence the result for proper morphisms. $\square$

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