Lemma 99.4.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be a surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times _\mathcal {Y} \mathcal{X} \to W$ has one of the separation properties of Definition 99.4.1 then so does $f$.

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 98.3.4. $\square$

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