The Stacks project

Proof. Assume $W \times _\mathcal {Y} \mathcal{X} \to W$ is quasi-compact. Let $\mathcal{Z} \to \mathcal{Y}$ be a morphism with $\mathcal{Z}$ a quasi-compact algebraic stack. Choose a scheme $U$ and a surjective smooth morphism $U \to W \times _\mathcal {Y} \mathcal{Z}$. Since $U \to \mathcal{Z}$ is flat, surjective, and locally of finite presentation and $\mathcal{Z}$ is quasi-compact, we can find a quasi-compact open subscheme $U' \subset U$ such that $U' \to \mathcal{Z}$ is surjective. Then $U' \times _\mathcal {Y} \mathcal{X} = U' \times _ W (W \times _\mathcal {Y} \mathcal{X})$ is quasi-compact by assumption and surjects onto $\mathcal{Z} \times _\mathcal {Y} \mathcal{X}$. Hence $\mathcal{Z} \times _\mathcal {Y} \mathcal{X}$ is quasi-compact as desired. $\square$