Lemma 88.9.2. 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

\[ (h : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V), \phi : h^*y \to f(x)\} /\cong \]is an algebraic space over $U$.

Here we have made a choice of pullbacks for $\mathcal{Y}$.

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