The Stacks project

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. $\square$