The Stacks project

Lemma 108.10.2. The morphism $\mathit{Mor}_ B(Y, X) \to B$ is separated and locally of finite presentation.

Proof. To check $\mathit{Mor}_ B(Y, X) \to B$ is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 108.10.1. The second statement is part of Quot, Proposition 99.12.3. $\square$

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