Lemma 14.3.9 (Adjunctions on \(\Oo \)-algebras). Every symmetric monoidal adjunction \(L\dashv R\) induces, for each \(\infty \)-operad \(\Oo \), an adjunction \[ \Alg _{\Oo }(L)\colon \Alg _{\Oo }(C) \rightleftarrows \Alg _{\Oo }(D)\noloc \Alg _{\Oo }(R). \] The underlying functors are obtained by applying \(L\) and \(R\), respectively, at every color of \(\Oo \).
Proof. By part (2) of Proposition 14.3.6, the adjunction \(L^{\otimes }\dashv R^{\otimes }\) is relative to \(\Span (\Fin )\). Applying Lemma 14.3.5 to \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) gives an adjunction \[ \Fun _{/\Span (\Fin )}(\Oo ^{\otimes },C^{\otimes }) \rightleftarrows \Fun _{/\Span (\Fin )}(\Oo ^{\otimes },D^{\otimes }). \] Postcomposition with both functors preserves finite-product-preserving functors because \(L^{\otimes }\) and \(R^{\otimes }\) are morphisms of \(\infty \)-operads. Restricting the adjunction to these subcategories gives the asserted adjunction on \(\Alg _{\Oo }(C)\) and \(\Alg _{\Oo }(D)\). □
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