Pointwise monoidal structure

Example 14.2.2 (Pointwise monoidal structure). If \(C\) is a symmetric monoidal \(\infty \)-category and \(I\) is an \(\infty \)-category, then \(\Fun (I,C)\) admits a symmetric monoidal structure, called the pointwise monoidal structure. It is given by the composite \[ \Span (\Fin ) \xrightarrow {(C,\otimes )} \Cat _{\infty } \xrightarrow {\Fun (I,-)} \Cat _{\infty }. \] This is functorial in both \(I\) and \(C\), resulting in a functor \[ \Fun (-,-)\colon \Cat _{\infty }\catop \times \Cat _{\infty }^{\otimes } \to \Cat _{\infty }^{\otimes }. \]