Proposition 18.5.4 (Multiplicative universal property of commutative monoids). Let \(C\) and \(D\) be presentably symmetric monoidal \(\infty \)-categories, and assume that \(D\) is semiadditive. Restriction along the free commutative monoid functor induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }\bigl (\CMon (C),D\bigr ) \iso \Fun ^{\mathrm {L},\otimes }(C,D) \] of \(\infty \)-categories.
Proof. Set \(M:=\CMon (\An )\). By Theorem 18.5.3, \(M\) is a mode and there is a natural equivalence \[ M\otimes C\simeq \CMon (C). \] Under this equivalence, the localization unit \(C\simeq \An \otimes C\to M\otimes C\) is the free semiadditive functor \(C\to \CMon (C)\). The symmetric monoidal refinement of this functor is the one constructed from Day convolution in Proposition 16.4.1.
Since \(D\) is semiadditive, it is \(M\)-local by Theorem 18.5.3. The localization \(M\otimes -\) of Proposition 18.5.2 is induced by the idempotent commutative algebra \(M\), so it restricts to a localization on commutative algebra objects of \(\PrL \). Its universal property gives an equivalence \[ \Fun ^{\mathrm {L},\otimes }(M\otimes C,D) \iso \Fun ^{\mathrm {L},\otimes }(C,D). \] Using the displayed identification of \(M\otimes C\) with \(\CMon (C)\) gives the result. Since the argument takes place in commutative algebra objects of \(\PrL \), it supplies the full symmetric monoidal coherence rather than only the individual tensor comparisons. □
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