Lemma 99.4.4. All of the separation axioms listed in Definition 99.4.1 are stable under base change.

Proof. Let $f : \mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y}' \to \mathcal{Y}$ be morphisms of algebraic stacks. Let $f' : \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ be the base change of $f$ by $\mathcal{Y}' \to \mathcal{Y}$. Then $\Delta _{f'}$ is the base change of $\Delta _ f$ by the morphism $\mathcal{X}' \times _{\mathcal{Y}'} \mathcal{X}' \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$, see Categories, Lemma 4.31.14. By the results of Properties of Stacks, Section 98.3 each of the properties of the diagonal used in Definition 99.4.1 is stable under base change. Hence the lemma is true. $\square$

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