The Stacks project

Proof. Note that $\Delta$ is a morphism of schemes 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 locally of finite type over $X$ (by Lemma 29.15.4). Thus the first statement holds by Lemma 29.21.11. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is a monomorphism, hence separated (Schemes, Lemma 26.23.3). $\square$