The Stacks project

Lemma 87.24.5. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces over $S$. If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$ is locally of finite type.

Proof. By Lemma 87.19.3 we see that $f$ is representable by algebraic spaces. Let $T$ be a scheme and let $T \to Z$ be a morphism. Then we can apply Morphisms of Spaces, Lemma 67.23.6 to the morphisms $T \times _ Z X \to T \times _ Z Y \to T$ of algebraic spaces to conclude. $\square$

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