The Stacks project

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 $f(x)$.