The Stacks project

Lemma 95.5.3. There exists a subcategory $\mathcal{QC}\! \mathit{oh}_{fg, small} \subset \mathcal{QC}\! \mathit{oh}_{fg}$ with the following properties:

  1. the inclusion functor $\mathcal{QC}\! \mathit{oh}_{fg, small} \to \mathcal{QC}\! \mathit{oh}_{fg}$ is fully faithful and essentially surjective, and

  2. the functor $p_{fg, small} : \mathcal{QC}\! \mathit{oh}_{fg, small} \to (\mathit{Sch}/S)_{fppf}$ turns $\mathcal{QC}\! \mathit{oh}_{fg, small}$ into a stack over $(\mathit{Sch}/S)_{fppf}$.

Proof. We have seen in Lemmas 95.5.1 and 95.5.2 that $p_{fg} : \mathcal{QC}\! \mathit{oh}_{fg} \to (\mathit{Sch}/S)_{fppf}$ satisfies (1), (2) and (3) of Stacks, Definition 8.4.1 as well as the additional condition (4) of Stacks, Remark 8.4.9. Hence we obtain $\mathcal{QC}\! \mathit{oh}_{fg, small}$ from the discussion in that remark. $\square$

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