Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim\_{j \in J} I\_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\varinjlim\_{i \in I} D\_i = \varinjlim\_{j \in J} \varinjlim\_{i \in I\_j} D\_i$. But I've never seen general conditions along these lines spelled out for 1-categories.


**Question 1:** Is there some place where conditions making the above true are given in the 1-categorical setting?


For $\infty$-categories, there is [Corollary 4.2.3.10](https://www.math.ias.edu/%7Elurie/papers/HTT.pdf#page=271) of Higher Topos Theory. Unfortunately, the formulation of the result is somewhat abstruse, being expressed in terms of the bespoke simplicial set denoted $K\_F$ there (defined using 4 conditions in Notation 4.2.3.1).


As a result, I'm having the following problem: it seems to me that for any cocone of $\infty$-categories $(I\_j \to I)\_{j \in J}$, one should be able to construct a natural map $\varinjlim\_{j \in J} \varinjlim\_{i \in I\_j} D\_i \to \varinjlim\_{i \in I} D\_i$, and one would expect HTT 4.2.3.10 to imply that under the appropriate conditions, *this* map is an equivalence. But the formulation doesn't seem to easily lend itself to confirming this.


**Question 2:** Is the natural map $\varinjlim\_{j \in J} \varinjlim\_{i \in I\_j} D\_i \to \varinjlim\_{i \in I} D\_i$ constructed somewhere in reasonable generality? (Or else is it easy to construct from general machinery given somewhere?)


**Question 3:** Is there written somewhere an account of conditions (perhaps analogous to those of HTT 4.2.3.10) which ensure that this map is an equivalence?



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
$$ \mathrm{colim}\_I \, D \simeq \mathrm{colim}\_E \, Dq \simeq \mathrm{colim}\_{j \in J} \, \mathrm{colim}\_{I\_j} \,D|\_{I\_j}. $$


