Proposition 6.164.
The adjunction \(\Cat \rightleftarrows \Cat^{\An\text{-}\colim}\) from Lemma 6.160 restricts to an adjunction
\[\Cat^{\wccolim} \rightleftarrows \Cat^{\colim}.\]
In particular, if \(C\) is a category with small weakly contractible colimits and \(D\) is a category with small colimits, then restriction along the inclusion \(C \hookrightarrow \int_{\An}C\) induces an equivalence
\[\Fun^{\colim}(\int_{\An}C, D) \iso \Fun^{\wccolim}(C,D).\]
Proof
We saw in Proposition 6.161 that restriction along the inclusion induces an equivalence
\[\Fun^{\An\text{-}\colim}(\int_{\An}C, D) \iso \Fun(C,D).\]
By part (1) of Proposition 6.162, this restriction functor restricts to a fully faithful functor \[\Fun^{\colim}(\int_{\An}C, D) \hookrightarrow \Fun^{\wccolim}(C,D).\]
For essential surjectivity, let \(F\colon C \to D\) be a functor preserving weakly contractible colimits, and consider its colimit-preserving extension \(\PSh^{\mathrm{small}}(C) \to D\). By part (4) of Proposition 6.162, this inverts the \(L\)-local maps in \(\PSh^{\mathrm{small}}(C)\), and so descends to a colimit-preserving functor \(\int_{\An}C \to D\) extending \(F\). This finishes the proof.