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 99.3 each of the properties of the diagonal used in Definition 100.4.1 is stable under base change. Hence the lemma is true. $\square$

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