Lemma 66.17.6. Having property $(\beta )$, being decent, or being reasonable is preserved under compositions.

**Proof.**
Let $\omega \in \{ \beta , decent, reasonable\} $. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over the scheme $S$. Assume $f$ and $g$ both have property $(\omega )$. Then we have to show that for any scheme $T$ and morphism $T \to Z$ the space $T \times _ Z X$ has $(\omega )$. By Lemma 66.17.4 this reduces us to the following claim: Suppose that $Y$ is an algebraic space having property $(\omega )$, and that $f : X \to Y$ is a morphism with $(\omega )$. Then $X$ has $(\omega )$. This is the content of Lemma 66.17.5.
$\square$

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