Lemma 94.9.2. Let S be a scheme contained in \mathit{Sch}_{fppf}. Let f : \mathcal{X} \to \mathcal{Y} be a 1-morphism of categories fibred in groupoids over (\mathit{Sch}/S)_{fppf}. The following are necessary and sufficient conditions for f to be representable by algebraic spaces:
for each scheme U/S the functor f_ U : \mathcal{X}_ U \longrightarrow \mathcal{Y}_ U between fibre categories is faithful, and
for each U and each y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U) the presheaf
(h : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V), \phi : h^*y \to f(x)\} /\congis an algebraic space over U.
Here we have made a choice of pullbacks for \mathcal{Y}.
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