Proposition 19.7.3. Let \(C\) be a small symmetric monoidal \(\infty \)-category which admits finite coproducts. Assume that for every object \(X\in C\), the functor \[ X\otimes -\colon C\longrightarrow C \] preserves finite coproducts. Then the cocartesian monoidal structure on \(C\) canonically refines \(C\) to an object of \[ \CRig (\Cat _{\infty })=\CAlg (\CMon (\Cat _{\infty })). \]
Proof. The finite-coproduct specialization in Section 22.4 exhibits the given symmetric monoidal structure on \(C\) as a commutative algebra object of \((\Cat _{\infty }^{\mathrm {coprod}},\otimes _{\mathrm {coprod}})\). Applying the lax symmetric monoidal functor of Proposition 19.7.2 produces a commutative algebra in \(\CMon (\Cat _{\infty })\), whose underlying commutative monoid is the cocartesian monoidal structure \((C,\amalg )\). □
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