Lemma 101.17.7. Let \mathcal{X} \to \mathcal{Y} \to \mathcal{Z} be morphisms of algebraic stacks. Assume \mathcal{X} \to \mathcal{Z} is locally of finite type and that \mathcal{X} \to \mathcal{Y} is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then \mathcal{Y} \to \mathcal{Z} is locally of finite type.
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 W \times _\mathcal {Z} \mathcal{Y}. Set U = V \times _\mathcal {Y} \mathcal{X} which is an algebraic space. We know that U \to V is surjective, flat, and locally of finite presentation and that U \to W is locally of finite type. Hence the lemma reduces to the case of morphisms of algebraic spaces. The case of morphisms of algebraic spaces is Descent on Spaces, Lemma 74.16.2. \square
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