The Stacks project

Remark 101.45.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $U \to \mathcal{X}$ be a morphism whose source is an algebraic space. Let $G \to H$ be the pullback of the morphism $\mathcal{I}_\mathcal {X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{I}_\mathcal {Y}$ to $U$. If $\Delta _ f$ is unramified, ├ętale, etc, so is $G \to H$. This is true because

\[ \xymatrix{ U \times _\mathcal {X} U \ar[r] \ar[d] & \mathcal{X} \ar[d]^{\Delta _ f} \\ U \times _\mathcal {Y} U \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]

is cartesian and the morphism $G \to H$ is the base change of the left vertical arrow by the diagonal $U \to U \times U$. Compare with the proof of Lemma 101.6.6.

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