Derived monoidal structure coherent

Construction 20.1.5. Let \((C,W,I)\) be an \(\infty \)-category with weak equivalences and cofibrations which comes equipped with a left derivable symmetric monoidal structure. For a finite set \(J\), let \(W^J\) denote the collection of morphisms in \(C^J \simeq C^{\otimes }_J\) given by \(J\)-tuples of weak equivalences, and let \(W^{\otimes }\) be the collection of morphisms in \(C^{\otimes }\) given by the union of the \(W^J\). By construction, the cocartesian fibration \(p_C\colon C^{\otimes } \to \Span (\Fin )\) inverts all morphisms in \(W^{\otimes }\), so the universal property of localizations produces a functor \[ p_{C[W^{-1}]}\colon C^{\otimes }[(W^{\otimes })^{-1}] \to \Span (\Fin ). \] The localization functor \(\gamma ^{\otimes }\colon C^{\otimes } \to C^{\otimes }[(W^{\otimes })^{-1}]\) then becomes a functor over \(\Span (\Fin )\).