Decomposing a (co)limit by decomposing the indexing diagram

Let $p \colon E \to J$ be the cocartesian fibration for the diagram $j \mapsto I_j$. Then the colimit over $E$ of $F \colon E \to C$ can always (assuming the appropriate colimits exist in $C$) be written as an iterated colimit: $$ \mathrm{colim}_E \, F \simeq \mathrm{colim}_J \, p_! F \simeq \mathrm{colim}_{j \in J} \, \mathrm{colim}_{I_j} \, F|_{I_j} $$ by first doing the colimit in two steps using the left Kan extension along $p$ and then that the inclusion $E_j \to E \times_J J_{/j}$ is cofinal since $p$ is cocartesian.

Now the colimit $I$ can be described as the localization of $E$ at the cocartesian morphisms. Since any localization is cofinal, this means there is a cofinal functor $q \colon E \to I$. For a functor $D \colon I \to C$, this means we have equivalences $$ \mathrm{colim}_I \, D \simeq \mathrm{colim}_E \, Dq \simeq \mathrm{colim}_{j \in J} \, \mathrm{colim}_{I_j} \,D|_{I_j}. $$

The way I always remember this stuff is as follows:

• Given a map $J \to \mathsf{Cat}$ form the associated cocartesian fibration $E \to J$.
• By assumption, $I$ is the actual colimit (as opposed to the left lax one) so we have a (weak) localization $E \to I$. Weak localizations are final (and initial, in fact) so, to compute the colimit over $I$ is the same as computing it over $E$.
• To compute the colimit over $E$ we may first left Kan extend to $J$.
• Since $E \to J$ is cocartesian, the map $E_x \to E_{/x}$ is final, and we may replace $E_{/x}$ with $E_x=I_x$ in the formula for left Kan extensions.

That gives the result.

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.