Example 7.6.5.8 (Sequential Colimits in $\operatorname{\mathcal{QC}}$). Suppose we are given a collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}(n) \} _{n \geq 0}$ and functors $F_ n: \operatorname{\mathcal{C}}(n) \rightarrow \operatorname{\mathcal{C}}(n+1)$, which we view as a diagram
\[ \operatorname{\mathcal{C}}(0) \xrightarrow { F_0 } \operatorname{\mathcal{C}}(1) \xrightarrow { F_1} \operatorname{\mathcal{C}}(2) \xrightarrow { F_2} \operatorname{\mathcal{C}}(3) \rightarrow \cdots \]
Let $\varinjlim _{n} \operatorname{\mathcal{C}}(n)$ denote the colimit of this diagram (formed in the ordinary category of simplicial sets). Then $\varinjlim _{n} \operatorname{\mathcal{C}}(n)$ is also an $\infty $-category, which is also a colimit of the associated diagram $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{QC}}$. This is a special case of Corollary 9.1.7.3.