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:

1. $\mathcal{X}$ is fibred in setoids1, and

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

Proof. Omitted, but see Categories, Lemma 4.40.2. $\square$

[1] This means that it is fibred in groupoids and objects in the fibre categories have no nontrivial automorphisms, see Categories, Definition 4.38.2.

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