Lemma 101.17.8. Let f : \mathcal{X} \to \mathcal{Y}, g : \mathcal{Y} \to \mathcal{Z} be morphisms of algebraic stacks. If g \circ f : \mathcal{X} \to \mathcal{Z} is locally of finite type, then f : \mathcal{X} \to \mathcal{Y} is locally of finite type.
Proof. We can find a diagram
\xymatrix{ U \ar[r] \ar[d] & V \ar[r] \ar[d] & W \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} \ar[r] & \mathcal{Z} }
where U, V, W are schemes, the vertical arrow W \to \mathcal{Z} is surjective and smooth, the arrow V \to \mathcal{Y} \times _\mathcal {Z} W is surjective and smooth, and the arrow U \to \mathcal{X} \times _\mathcal {Y} V is surjective and smooth. Then also U \to \mathcal{X} \times _\mathcal {Z} V is surjective and smooth (as a composition of a surjective and smooth morphism with a base change of such). By definition we see that U \to W is locally of finite type. Hence U \to V is locally of finite type by Morphisms, Lemma 29.15.8 which in turn means (by definition) that \mathcal{X} \to \mathcal{Y} is locally of finite type. \square
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