Definition 6.165.
Let \(T\) be a cocomplete category and let \(C\) be a category admitting weakly contractible colimits. We define the category of \(T\)-parametrized objects of \(C\) as the tensor product
\[\int_T C \quad := \quad T \otimes \int_{\An} C\]
in the category \(\Cat^{\colim}\) of cocomplete categories (using that \(\int_{\An}C\) admits colimits by Proposition 6.162).