Corollary 6.166.
The functor \(T \times C \hookrightarrow T \times \int_{\An} C \to \int_T C\) is universal among functors \(F\colon T \times C \to D\) into a cocomplete category \(D\) which preserve all colimits in the first variable and weakly contractible colimits in the second variable.
Proof
By definition, the tensor product \(\int_TC = T \otimes \int_{\An} C\) comes equipped with a functor from \(T \times \int_{\An}C\) satisfying the universal property that for every other cocomplete category \(D\), precomposition with this functor defines an equivalence
\[\Fun^{\colim}(\int_TC, D) \iso \Fun^{\colim,\colim}(T \times \int_{\An} C,D) \iso \Fun^{\colim}(T, \Fun^{\colim}(\int_{\An}C,D)).\]
The claimed universal property for \(\int_T C\) thus follows immediately from the universal property of \(\int_{\An}C\) established in Proposition 6.164.