The Stacks project

Proof. The category $\mathcal{C}$ is the kernel of the exact functor $v : \textit{Adeq}(\mathcal{O}) \to \mathit{QCoh}(\mathcal{O}_ S)$, see Lemma 46.6.1. Hence it is a Serre subcategory by Homology, Lemma 12.10.4. By Homology, Lemma 12.10.6 we obtain an induced exact functor $\overline{v} : \textit{Adeq}(\mathcal{O}) / \mathcal{C} \to \mathit{QCoh}(\mathcal{O}_ S)$. Because $u$ is a right inverse to $v$ we see right away that $\overline{v}$ is essentially surjective. We see that $\overline{v}$ is faithful by Homology, Lemma 12.10.7. Because $u$ is a right inverse to $v$ we finally conclude that $\overline{v}$ is fully faithful. $\square$