Corollary 14.5.4 (cf. [Lurie (2017), Proposition 2.2.1.9]). Let \(L \colon C \rightleftarrows D \colon i\) be as in Proposition 14.5.3, and let \(\Oo \) be an \(\infty \)-operad. Then postcomposition with the symmetric monoidal functors \(i^{\otimes }\) and \(L^{\otimes }\) induces an adjunction \[ \Alg _{\Oo }(L)\colon \Alg _{\Oo }(C) \rightleftarrows \Alg _{\Oo }(D) \colon \Alg _{\Oo }(i) \] in which the right adjoint is fully faithful. Its essential image consists of those \(\Oo \)-algebras \(A\) in \(C\) whose underlying objects \(A(o) \in C\) lie in \(D\), for every color \(o \in \Oo ^{\simeq }\). The left adjoint is computed by \(L\) on underlying objects, so that \(\Alg _{\Oo }(D)\) is the Bousfield localization of \(\Alg _{\Oo }(C)\) at those maps of \(\Oo \)-algebras whose underlying morphisms are \(L\)-equivalences.
Proof. By part (1) of Proposition 14.5.3, the symmetric monoidal \(\infty \)-category \(D^{\otimes }\) is a full suboperad of \(C^{\otimes }\). Hence postcomposition with \(i^{\otimes }\) is fully faithful, and an \(\infty \)-operad map \(\Oo ^{\otimes } \to C^{\otimes }\) factors through \(D^{\otimes }\) if and only if it does so on the colors: an object of \(C^{\otimes }\) lying over \(\lra {n}\) decomposes, via the inert morphisms, into an \(n\)-tuple of objects of \(C\), and an \(\infty \)-operad map preserves inert morphisms. This gives full faithfulness of \(\Alg _{\Oo }(i)\) together with the description of the essential image.
By part (2) of Proposition 14.5.3, the adjunction \(L^{\otimes }\dashv i^{\otimes }\) is a symmetric monoidal adjunction. Applying Lemma 14.3.9 gives the asserted adjunction on \(\Oo \)-algebras. The unit is computed colorwise by the unit of \(L\dashv i\), so the left adjoint is computed by \(L\) on underlying objects and inverts precisely the maps whose underlying morphisms are \(L\)-equivalences. □
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