Lemma 6.159.
Let \(D\) be a category admitting anima-indexed colimits. Then the inclusion \(D \hookrightarrow \int_{\An}D\) admits a left adjoint \(\colim \colon \int_{\An} D \to D\), given on objects by sending \((A,X)\) to \(\colim_{a \in A} X_a\).
Proof
Since left adjoints may be constructed objectwise, it will suffice to show that for every \(Y \in D\) there is a natural equivalence
\[\Hom_{\int_{\An}D}((A,X), (*,Y)) \iso \Hom_{D}(\colim_{a \in A} X_a,Y).\]
But this is clear: since there is a unique map \(A \to *\) in \(\An\), the left-hand side simplifies to \(\Hom_{\Fun(A,D)}(X, \const_Y)\), which by the very definition of colimits in \(D\) is also the right-hand side.