Lemma 101.19.4. 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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