Lemma 99.19.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ be a point. The following are equivalent

for some morphism $x : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ in the class of $x$ setting $y = f \circ x$ the map $G_ x \to G_ y$ of automorphism group algebraic spaces is an isomorphism, and

for any morphism $x : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ in the class of $x$ setting $y = f \circ x$ the map $G_ x \to G_ y$ of automorphism group algebraic spaces is an isomorphism.

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