Left adjoint to the map into a limit of slices

Proof. The colimit functor \(\colim \colon \Fun (I,T) \to T\) admits a right adjoint \(\const \colon T \to \Fun (I,T)\) sending an object \(Y\) to the constant functor on \(Y\). By slicing this adjunction over an arbitrary diagram \(X_{\bullet } \in \Fun (I,T)\), with colimit denoted \(X\), we obtain another adjunction \[ \colim \colon \Fun (I,T)_{/X_{\bullet }} \rightleftarrows T_{/X}, \] see [Lurie (2009), Proposition 5.2.5.1]. The right adjoint in this adjunction is given by the composite \(T_{/X} \xrightarrow {\const } \Fun (I,T)_{/\const X} \to \Fun (I,T)_{/X_{\bullet }}\), with the second map given by base change along the colimit cocone \(X_{\bullet } \to \const _X\). We now make the following two observations:

• The \(\infty \)-category \(\Fun (I,T)_{/X_{\bullet }}\) contains the limit \(\lim _{i \in I\catop } C_{/X_i} \simeq \Fun ^{\cart }(I,T)_{/X_{\bullet }}\) as a full subcategory: given transformations \(Y_{\bullet } \to Z_{\bullet } \to X_{\bullet }\), if both \(Z_{\bullet } \to X_{\bullet }\) and \(Y_{\bullet } \to X_{\bullet }\) are cartesian transformations, then so is \(Y_{\bullet } \to Z_{\bullet }\) by the pasting law for pullback squares.
• Since any transformation \(\const Y \to \const X\) between constant diagrams is cartesian, and cartesian transformations are closed under base change, the right adjoint \(T_{/X} \to \Fun (I,T)_{/X_{\bullet }}\) lands in this full subcategory.

It follows that the adjunction restricts to an adjunction \(\lim _i T_{/X_i} \rightleftarrows T_{/X}\), as desired. □