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, étale, 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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