The Stacks project

Remark 101.16.4. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and stable under base change. Then the property of morphisms of algebraic stacks defined in Definition 101.16.2 is stable under base change. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y}' \to \mathcal{Y}$ be morphisms of algebraic stacks and assume $f$ has property $\mathcal{P}$. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective smooth morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \to \mathcal{Y}' \times _\mathcal {Y} V$. Then the morphism $U \to V$ has property $\mathcal{P}$ by definition. Whence $V' \times _ V U \to V'$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under base change. Considering the diagram

we see that the left top horizontal arrow is smooth and surjective, whence by definition we see that the projection $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ has property $\mathcal{P}$.