Remark 99.19.5. 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 99.19.4 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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