Corollary 14.1.21. The functor \(\Triv _{(-)}\colon \Cat _{\infty } \to \Op _{\infty }\) is left adjoint to the functor \[ (-)_{\lra {1}}\colon \Op _{\infty } \to \Cat _{\infty }, \qquad \Oo \mapsto \Oo _{\lra {1}}, \] and it is fully faithful.
Proof. The equivalence of Lemma 14.1.20 is given by passing to underlying \(\infty \)-categories and precomposing with the isomorphism \(C \iso (\Triv _C)_{\lra {1}}\), and is thus natural in both variables. Passing to groupoid cores yields the corresponding equivalence of mapping animae, so it exhibits \(\Triv _{(-)}\) as a left adjoint of \((-)_{\lra {1}}\). Taking \(\Oo = \Triv _D\) for an \(\infty \)-category \(D\), the equivalence \[ \Fun _{\Op _{\infty }}(\Triv _C,\Triv _D) \iso \Fun (C,(\Triv _D)_{\lra {1}}) \simeq \Fun (C,D) \] is inverse to the map induced by \(\Triv _{(-)}\), which is therefore fully faithful. □
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