Lemma 95.8.2. There exists a subcategory \mathcal{S}\! \mathit{paces}_{ft, small} \subset \mathcal{S}\! \mathit{paces}_{ft} with the following properties:
the inclusion functor \mathcal{S}\! \mathit{paces}_{ft, small} \to \mathcal{S}\! \mathit{paces}_{ft} is fully faithful and essentially surjective, and
the functor p_{ft, small} : \mathcal{S}\! \mathit{paces}_{ft, small} \to (\mathit{Sch}/S)_{fppf} turns \mathcal{S}\! \mathit{paces}_{ft, small} into a stack over (\mathit{Sch}/S)_{fppf}.
Proof. We have seen in Lemmas 95.8.1 that p_{ft} : \mathcal{S}\! \mathit{paces}_{ft} \to (\mathit{Sch}/S)_{fppf} satisfies (1), (2) and (3) of Stacks, Definition 8.4.1. The additional condition (4) of Stacks, Remark 8.4.9 holds because every algebraic space X over S is of the form U/R for U, R \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}), see Spaces, Lemma 65.9.1. Thus there is only a set worth of isomorphism classes of objects. Hence we obtain \mathcal{S}\! \mathit{paces}_{ft, small} from the discussion in that remark. \square
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