Corollary 7.4.4.17. Let $K$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram of $\infty $-categories satisfying the following conditions:
- $(a)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ admits $K$-indexed colimits.
- $(b)$
For each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}(e): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ preserves $K$-indexed colimits.
Then the $\infty $-category $\varprojlim (\mathscr {F})$ admits $K$-indexed colimits. Moreover, a map $\overline{q}: K^{\triangleright } \rightarrow \varprojlim (\mathscr {F})$ is a colimit diagram if and only if its image in $\mathscr {F}(C)$ is a colimit diagram, for each $C \in \operatorname{\mathcal{C}}$.