Lemma 100.10.2. Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be an integral (or finite) morphism of algebraic stacks. Then $\mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X}$ is an integral (or finite) morphism of algebraic stacks.

**Proof.**
This follows from the discussion in Properties of Stacks, Section 99.3.
$\square$

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