Lemma 66.25.5. Let $S$ be a scheme. Let $X$ be a decent, locally Noetherian, and universally catenary algebraic space over $S$. Then any decent algebraic space locally of finite type over $X$ is universally catenary.

**Proof.**
This is formal from the definitions and the fact that compositions of morphisms locally of finite type are locally of finite type (Morphisms of Spaces, Lemma 65.23.2).
$\square$

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