Lemma 99.27.6. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks If $g \circ f$ is locally of finite presentation and $g$ is locally of finite type, then $f$ is locally of finite presentation.

Proof. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y} \times _\mathcal {Z} W$. Choose an algebraic space $U$ and a surjective smooth morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. The lemma follows upon applying Morphisms of Spaces, Lemma 65.28.9 to the morphisms $U \to V \to W$. $\square$

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