The Stacks project

Proof. To check $\mathcal{C}\! \mathit{oh}_{X/B} \to B$ is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 108.4.1. To prove that $\mathcal{C}\! \mathit{oh}_{X/B} \to B$ is locally of finite presentation, we have to show that $\mathcal{C}\! \mathit{oh}_{X/B} \to B$ is limit preserving, see Limits of Stacks, Proposition 102.3.8. This follows from Quot, Lemma 99.5.6 (small detail omitted). $\square$