The Stacks project

Proof. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite presentation. This means there exists a diagram as in Lemma 67.22.1 with $h$ locally of finite presentation and surjective vertical arrow $a$. By Morphisms, Lemma 29.21.8 $h$ is locally of finite type. Hence $X \to Y$ is locally of finite type by definition. If $f$ is of finite presentation then it is quasi-compact and it follows that $f$ is of finite type. $\square$