Lemma 90.10.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ be categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be a property as in Definition 90.10.1 which is stable under composition. Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms which are representable by algebraic spaces. If $f$ and $g$ have property $\mathcal{P}$ so does $g \circ f : \mathcal{X} \to \mathcal{Z}$.

**Proof.**
Note that the lemma makes sense by Lemma 90.9.9. Proof omitted.
$\square$

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