Lemma 93.10.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $a : F \to G$ be a map of presheaves on $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be as in Definition 93.10.1. Assume $a$ is representable by algebraic spaces. Then $a : F \to G$ has property $\mathcal{P}$ (see Bootstrap, Definition 79.4.1) if and only if the corresponding morphism $\mathcal{S}_ F \to \mathcal{S}_ G$ of categories fibred in groupoids has property $\mathcal{P}$.

**Proof.**
Note that the lemma makes sense by Lemma 93.9.5. Proof omitted.
$\square$

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