The Stacks project

Proof. Let $j : X \to \overline{X}$ be a compactification of $X$ over $S$. Then we let $\overline{X}'$ be the scheme theoretic image of $(j, f) : X \to \overline{X} \times _ S Y$. The morphism $\overline{X}' \to Y$ is proper because $\overline{X} \times _ S Y \to Y$ is proper as a base change of $\overline{X} \to S$. On the other hand, since $Y$ is separated over $S$, the morphism $(1, f) : X \to X \times _ S Y$ is a closed immersion (Schemes, Lemma 26.21.10) and hence $X \to \overline{X}'$ is an open immersion by Morphisms, Lemma 29.6.8 applied to the “partial section” $s = (j, f)$ to the projection $\overline{X} \times _ S Y \to \overline{X}$. $\square$