Lemma 89.19.6. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. Then $\mathit{Aut}(x): \mathcal{C}_ A \to \textit{Sets}$ satisfies (RS).

Proof. It follows that $\mathit{Aut}(x)$ satisfies (RS) from the fully faithfulness of the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ in Lemma 89.16.4. $\square$

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