The Stacks project

Lemma 90.10.4. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Let $\mathcal{P}$ be as in Definition 90.10.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in setoids over $(\mathit{Sch}/S)_{fppf}$. Let $F$, resp. $G$ be the presheaf which to $T$ associates the set of isomorphism classes of objects of $\mathcal{X}_ T$, resp. $\mathcal{Y}_ T$. Let $a : F \to G$ be the map of presheaves corresponding to $f$. Then $a$ has $\mathcal{P}$ if and only if $f$ has $\mathcal{P}$.

Proof. The lemma makes sense by Lemma 90.9.6. The lemma follows on combining Lemmas 90.10.2 and 90.10.3. $\square$


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