Proposition 14.4.7 (cf. [Lurie (2017), Corollary 7.3.2.7]). Let \(L\colon C \to D\) be an \(\Oo \)-monoidal functor between \(\Oo \)-monoidal \(\infty \)-categories. Assume that for every color \(o \in \Oo ^{\simeq }\), the functor \(L_o\colon C_o \to D_o\) admits a right adjoint \(R_o\colon D_o \to C_o\). Then the functor \(L^{\otimes }\colon C^{\otimes } \to D^{\otimes }\) admits a right adjoint \(R^{\otimes }\colon D^{\otimes } \to C^{\otimes }\). The functor \(R^{\otimes }\) has a canonical structure over \(\Oo ^{\otimes }\) for which \(L^{\otimes }\dashv R^{\otimes }\) is a relative adjunction, and with this structure it is a morphism of operads over \(\Oo \). In particular, \(L\) admits a lax \(\Oo \)-monoidal right adjoint \(R\).
Proof. The proof is entirely analogous to that of Proposition 14.3.6 and is left to the reader. □
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