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 of Higher Topos Theory. Unfortunately, the formulation of the result is somewhat abstruse, being formulated in terms of the bespoke simplicial set denoted $K_F$ there (defined 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?

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?

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 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?