The Stacks project

Lemma 66.17.7. Let $S$ be a scheme. Let $f : X \to Y$, $g : Z \to Y$ be morphisms of algebraic spaces over $S$. If $X$ and $Z$ are decent (resp. reasonable, resp. have property $(\beta )$ of Lemma 66.5.1), then so does $X \times _ Y Z$.

Proof. Namely, by Lemma 66.17.3 the morphism $X \to Y$ has the property. Then the base change $X \times _ Y Z \to Z$ has the property by Lemma 66.17.4. And finally this implies $X \times _ Y Z$ has the property by Lemma 66.17.5. $\square$

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