Let $\lambda_j : \mathcal{I}_j \to \mathcal{I}$ be the component of the colimit cocone in $\textbf{Cat}$. Then, $$\textstyle [I, \mathcal{C}](D, \Delta T) \cong \varprojlim_\mathcal{J} [\mathcal{I} j, \mathcal{C}](D \lambda_j, \Delta T)$$ so if the relevant colimits exist in $\mathcal{C}$, $$\textstyle \mathcal{C} \left( \varinjlim_\mathcal{I} D, T \right) \cong \varprojlim_\mathcal{J} \mathcal{C} \left( \varinjlim_{\mathcal{I}_j} D \lambda_j, T \right) \cong \mathcal{C} \left( \varinjlim_\mathcal{J} \varinjlim_{\mathcal{I}_j} D \lambda_j, T \right)$$ as desired.

I assume $\varinjlim_{j : \mathcal{J}} \mathcal{I}_j = \mathcal{I}$ is meant in the strict sense of 1-categories. Since $\textbf{Cat}$ is cartesian closed, $$\textstyle [\mathcal{I}, \mathcal{C}] \cong \varprojlim_{j : \mathcal{J}} [\mathcal{I}_j, \mathcal{C}]$$ where the limit on the RHS is also meant in the strict sense of 1-categories. Let $\lambda_j : \mathcal{I} j \to \mathcal{I}$ be the component of the colimit cocone in $\textbf{Cat}$. Then, we also get a limit formula for the hom-sets of $[\mathcal{I}, \mathcal{C}]$, namely, $$\textstyle [\mathcal{I}, \mathcal{C}](D, \Delta T) \cong \varprojlim_{j : \mathcal{J}} [\mathcal{I}_j, \mathcal{C}](D \lambda_j, \Delta T)$$ so if the relevant colimits exist in $\mathcal{C}$, $$\textstyle \mathcal{C} \left( \varinjlim_\mathcal{I} D, T \right) \cong \varprojlim_{j : \mathcal{J}} \mathcal{C} \left( \varinjlim_{\mathcal{I}_j} D \lambda_j, T \right) \cong \mathcal{C} \left( \varinjlim_\mathcal{J} \varinjlim_{\mathcal{I}_j} D \lambda_j, T \right)$$ as desired.