Corollary 18.4.7. Let \(C\) be a symmetric monoidal \(\infty \)-category with finite limits, and assume that \(\oSp (\Mm _C)\) is represented by a symmetric monoidal structure on \(\Sp (C)\). Then \(\Omega ^{\infty }\colon \Sp (C) \to C\) admits a canonical lax symmetric monoidal structure. Furthermore, for every stably symmetric monoidal \(\infty \)-category \(D\), composition with \(\Omega ^{\infty }\) induces an equivalence of \(\infty \)-categories \[ \Omega ^{\infty } \circ - \colon \Fun ^{\otimes \dlax ,\lex }(D,\Sp (C)) \iso \Fun ^{\otimes \dlax ,\lex }(D,C), \] where the superscript \(\lex \) denotes the full subcategories of lax symmetric monoidal functors whose underlying functors preserve finite limits. On the left-hand side this is equivalently the condition that the underlying functor be exact.
Proof. Applying the adjunction of Theorem 18.4.6 to finite-operadic-limit-preserving maps \(\Mm _D\to \Mm _C\) gives the asserted equivalence on maximal animae. The enhancement to an equivalence of \(\infty \)-categories, including non-invertible lax monoidal natural transformations, is [Nikolaus (2016), Corollary 4.13]; it is obtained by applying the functoriality of operadic stabilization to the corresponding operad-map categories. □
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