Lemma 37.64.4. In Situation 37.64.1. If $X$ and $Y$ are separated, then the pushout $Y \amalg _ Z X$ (Proposition 37.64.3) is separated. Same with “separated over $S$”, “quasi-separated”, and “quasi-separated over $S$”.
Proof. The morphism $Y \amalg X \to Y \amalg _ Z X$ is surjective and universall closed. Thus we may apply Morphisms, Lemma 29.41.11. $\square$
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