Lemma 101.4.4. All of the separation axioms listed in Definition 101.4.1 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 100.3 each of the properties of the diagonal used in Definition 101.4.1 is stable under base change. Hence the lemma is true.
$\square$

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