$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.1.9.9. Suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_3 \xrightarrow { F_2 } \operatorname{\mathcal{C}}_2 \xrightarrow { F_1 } \operatorname{\mathcal{C}}_1 \xrightarrow { F_0 } \operatorname{\mathcal{C}}_0, \]
where each of the functors $F_{n}$ is an isofibration. Let $q: K \rightarrow \varprojlim _{n} \operatorname{\mathcal{C}}_{n}$ be a diagram, which we identify with a compatible sequence of diagrams $\{ q_ n: K \rightarrow \operatorname{\mathcal{C}}_ n \} _{n \geq 0}$. Assume that:
For each $n \geq 0$, the diagram $q_{n}$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_ n$.
For each $n \geq 0$, the functor $F_{n}: \operatorname{\mathcal{C}}_{n+1} \rightarrow \operatorname{\mathcal{C}}_ n$ preserves the colimit of the diagram $q_{n+1}$.
Then $q$ admits a colimit in the $\infty $-categoruy $\varprojlim _{n} \operatorname{\mathcal{C}}_ n$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \varprojlim _{n} \operatorname{\mathcal{C}}_ n$ is a colimit diagram if and only if its image in each $\operatorname{\mathcal{C}}_{n}$ is a colimit diagram.
Proof.
Set $\operatorname{\mathcal{C}}= \prod _{n \geq 0} \operatorname{\mathcal{C}}_ n$ and let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the shift functor (given on objects by the construction $\{ C_ n \} _{n \geq 0} \mapsto \{ F_ n( C_{n+1} ) \} _{n \geq 0}$). It follows from Corollary 4.5.6.19 that the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim _{n} \operatorname{\mathcal{C}}_ n \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ (\operatorname{id}, \operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id}, S) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}. } \]
is a categorical pullback square. The desired result now follows by combining Proposition 7.1.9.8 with Example 7.1.3.11.
$\square$