The Stacks project

Proof. We apply Criteria for Representability, Lemma 97.5.6. Let $V$ be an affine scheme $V$ locally of finite presentation over $S$ and let $\theta $ be an object of $\mathcal{X} \times \mathcal{X}$ over $V$. Let $F_\theta $ be an algebraic space representing $\mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, \theta } (\mathit{Sch}/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism (see Algebraic Stacks, Section 94.9). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas 98.11.2 and 98.11.3 we see that $F_\theta \to S$ is locally of finite presentation. It follows that $F_\theta \to V$ is locally of finite type by Morphisms of Spaces, Lemma 67.23.6. $\square$