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?

thatsubtle in the $1$-categorical case; I think it mostly relies on the fact that Cat is cartesian closed, and on the analysis of hom-sets in alimitof categories) $\endgroup$6more comments