Corollary 17.3.14. Let \(\Oo \) be an \(\infty \)-operad and let \(C\) be a symmetric monoidal \(\infty \)-category. There is a natural equivalence \[ \Alg _{\Oo }(C) \simeq \Fun ^{\otimes }(\Env (\Oo ),C). \] In other words, an \(\Oo \)-algebra in \(C\) is equivalently a symmetric monoidal functor from the envelope of \(\Oo \) to \(C\).
Proof. Let \(E\) be an \(\infty \)-category. Using Lemma 14.2.12, the adjunction of Proposition 17.3.13, and currying, we obtain natural equivalences \begin {align*} \Hom _{\Cat _{\infty }}(E,\Alg _{\Oo }(C)) &\simeq \Hom _{\Op _\infty }(\Oo ,\Mm _{\Fun (E,C)}) \\ &\simeq \Map ^\otimes (\Env (\Oo ),\Fun (E,C)) \\ &\simeq \Hom _{\Cat _{\infty }}(E,\Fun ^\otimes (\Env (\Oo ),C)). \end {align*}
Here \(\Fun (E,C)\) carries the pointwise symmetric monoidal structure. Since these equivalences are natural in \(E\), the Yoneda lemma gives the claimed equivalence of \(\infty \)-categories. □
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