Lemma 94.10.8. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{P}$ be a property as in Definition 94.10.1 which is stable under composition. Let $\mathcal{X}_ i, \mathcal{Y}_ i$ be categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, $i = 1, 2$. Let $f_ i : \mathcal{X}_ i \to \mathcal{Y}_ i$, $i = 1, 2$ be $1$-morphisms representable by algebraic spaces. If $f_1$ and $f_2$ have property $\mathcal{P}$ so does $ f_1 \times f_2 : \mathcal{X}_1 \times \mathcal{X}_2 \to \mathcal{Y}_1 \times \mathcal{Y}_2 $.
Proof. The lemma makes sense by Lemma 94.9.10. Proof omitted. $\square$
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