The Stacks project

Proof. Denote $g : W \times _\mathcal {Y} \mathcal{X} \to W$ the base change. Then $\Delta _ g$ is the base change of $\Delta _ f$ by the morphism $q : W \times _\mathcal {Y} (\mathcal{X} \times _\mathcal {Y} \mathcal{X}) \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Since $q$ is the base change of $W \to \mathcal{Y}$ we see that $q$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Hence the result follows from Properties of Stacks, Lemma 100.3.4. $\square$