Example 6.181.
Let \(K'\) be the class of all small categories and let \(K\) be the class of the weakly contractible categories. If \(C\) is a category with weakly contractible colimits, the category \(\Pp^{\all}_{\mathrm{wc}}(C)\) is universal among functors \(C \to C'\) into a cocomplete category \(C'\) which preserve weakly contractible colimits. Since this is also precisely the universal property of the category \(\int_{\An} C\) established in Proposition 6.164, we obtain an equivalence
\[\int_{\An}C \iso \Pp^{\all}_{\mathrm{wc}}(C).\]