The left adjoint to the comparison of slices

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 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 category \(\Fun(I,T)_{/X_{\bullet}}\) contains the limit \(\lim_{i \in I\catop} T_{/X_i} \simeq \Fun^{\cart}(I,T)_{/X_{\bullet}}\) 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.