Corollary 7.1.9.10. Let $K$ be a simplicial set and suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_3 \xrightarrow { F_2 } \operatorname{\mathcal{C}}_2 \xrightarrow { F_1 } \operatorname{\mathcal{C}}_1 \xrightarrow { F_0 } \operatorname{\mathcal{C}}_0. \]
Assume that each of the $\infty $-categories $\operatorname{\mathcal{C}}_ n$ admits $K$-indexed colimits and that each of the functors $F_{n}$ is an isofibration which preserves $K$-indexed colimits. Then:
The $\infty $-category $\varprojlim _{n} \operatorname{\mathcal{C}}_ n$ admits $K$-indexed colimits.
A morphism $\overline{q}: K^{\triangleright } \rightarrow \varprojlim _{n} \operatorname{\mathcal{C}}_ n$ is a colimit diagram if and only if its image in each $\operatorname{\mathcal{C}}_{n}$ is a colimit diagram.