Lemma 65.28.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ with diagonal $\Delta : X \to X \times _ Y X$. If $f$ is locally of finite type then $\Delta $ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta $ is of finite presentation.

**Proof.**
Note that $\Delta $ is a morphism over $X$ (via the second projection $X \times _ Y X \to X$). Assume $f$ is locally of finite type. Note that $X$ is of finite presentation over $X$ and $X \times _ Y X$ is of finite type over $X$ (by Lemma 65.23.3). Thus the first statement holds by Lemma 65.28.9. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is separated (Lemma 65.4.1).
$\square$

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