Remark 101.19.5. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Let x \in |\mathcal{X}| be a point. To indicate the equivalent conditions of Lemma 101.19.4 are satisfied for f and x in the literature the terminology f is stabilizer preserving at x or f is fixed-point reflecting at x is used. We prefer to say f induces an isomorphism between automorphism groups at x and f(x).
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