Lemma 92.8.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Then $\mathcal{X}$ is representable by an algebraic space over $S$ if and only if the following conditions are satisfied:

$\mathcal{X}$ is fibred in setoids

^{1}, andthe presheaf $U \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)/\! \! \cong $ is an algebraic space.

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