Algebras in pointwise monoidal structure

Proof. The pointwise symmetric monoidal structure of Example 14.2.2 is classified by the composite \[ \Span (\Fin )\xrightarrow {\Str ^{\cc }(p_C)} \Cat _{\infty }\xrightarrow {\Fun (I,-)}\Cat _{\infty }. \] By Proposition 23.2.4, its total category is \(\Span (\Fin )\times _{\Fun (I,\Span (\Fin ))} \Fun (I,C^{\otimes })\). Consequently, before imposing the finite-product condition, a functor over \(\Span (\Fin )\) from \(\Oo ^{\otimes }\) to this total category amounts to a functor \[ \Oo ^{\otimes }\longrightarrow \Fun (I,C^{\otimes }) \] whose composite with \(\Fun (I,p_C)\) is the constant \(I\)-diagram on \(p_{\Oo }\). Currying identifies these with functors \[ I\longrightarrow \Fun _{/\Span (\Fin )}(\Oo ^{\otimes },C^{\otimes }). \] Under this identification, preservation of finite products is pointwise in \(I\): the functor into the pointwise multimorphism operad is an operad map precisely when the corresponding functor \(\Oo ^{\otimes }\to C^{\otimes }\) is an operad map at every object of \(I\). Thus the equivalence restricts to the asserted equivalence of algebra categories. □