The Stacks project

Lemma 108.5.2. The morphism $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$ is separated. If $\mathcal{F}$ is of finite presentation, then it is also locally of finite presentation.

Proof. To check $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$ is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 108.5.1. The second statement is part of Quot, Proposition 99.8.4. $\square$

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