Proposition 6.161.
Let \(C\) and \(D\) be categories and assume that \(D\) admits anima-indexed colimits. Then restriction along \(i\colon C \hookrightarrow \int_{\An}C\) induces an equivalence
\[\Fun^{\An\text{-}\colim}(\int_{\An}C,D) \iso \Fun(C,D).\]
Proof
By the Yoneda lemma, we may check this after applying \(\Hom_{\Cat}(E,-)\) for all \(E \in \Cat\). Observe that \(\Fun(E,D)\) still admits anima-indexed colimits, which are computed pointwise in \(D\). Under currying-uncurrying, the map in question then translates to the map
\[\Hom_{\Cat^{\An\text{-}\colim}}(\int_{\An}C, \Fun(E,D)) \to \Hom_{\Cat}(C,\Fun(E,D))\]
given by restriction along \(i \colon C \hookrightarrow \int_{\An}C\). This is an isomorphism by the adjunction from the previous lemma.