The Stacks project

Lemma 101.14.8. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times _\mathcal {Y} \mathcal{X} \to W$ is universally injective, then $f$ is universally injective.

Proof. Observe that the diagonal $\Delta _ g$ of the morphism $g : W \times _\mathcal {Y} \mathcal{X} \to W$ is the base change of $\Delta _ f$ by $W \to \mathcal{Y}$. Hence if $\Delta _ g$ is surjective, then so is $\Delta _ f$ by Properties of Stacks, Lemma 100.3.3. Thus the lemma follows from the characterization (2) in Lemma 101.14.5. $\square$

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