Corollary 15.2.17. The functor \(\OpCocart \colon \Cat _{\infty } \to \Op _{\infty }\) is fully faithful and induces an equivalence \[ \Cat _{\infty } \iso \Op ^{\cocart }_{\infty }. \]
Proof. Observe that \(\OpCocart \colon \Cat _{\infty } \to \Op _{\infty }\) admits a retraction, given by \((-)_{\lra {1}}\colon \Op _{\infty } \to \Cat _{\infty }\). It follows that for two \(\infty \)-categories \(C\) and \(D\), the composite \[ \Hom _{\Cat _{\infty }}(C,D) \to \Hom _{\Op _{\infty }}(\OpCocart _C,\OpCocart _D) \to \Hom _{\Cat _{\infty }}(C,D) \] is the identity. Since the second map is an equivalence by Proposition 15.2.16, the first map is an equivalence as well, showing full faithfulness. Since \(\Op ^{\cocart }_{\infty }\) is by definition the image of \(\OpCocart \), the second claim follows immediately. □
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