The Stacks project

Remark 101.34.3. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target and stable under composition. Then the property of DM morphisms of algebraic stacks defined in Definition 101.34.2 is stable under composition. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be DM morphisms of algebraic stacks having property $\mathcal{P}$. By Lemma 101.4.10 the composition $g \circ f$ is DM. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose an algebraic space $V$ and a surjective étale morphism $V \to \mathcal{Y} \times _\mathcal {Z} W$ (Lemma 101.21.7). Choose an algebraic space $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. Then the morphisms $V \to W$ and $U \to V$ have property $\mathcal{P}$ by definition. Whence $U \to W$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under composition. Thus, by definition again, we see that $g \circ f : \mathcal{X} \to \mathcal{Z}$ has property $\mathcal{P}$.