Corollary 4.5.7.19 (Sequential Limits as Pullbacks). Suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_3 \xrightarrow { F_2 } \operatorname{\mathcal{C}}_2 \xrightarrow { F_1 } \operatorname{\mathcal{C}}_1 \xrightarrow { F_0 } \operatorname{\mathcal{C}}_0. \]
Set $\operatorname{\mathcal{C}}= \prod _{n \geq 0} \operatorname{\mathcal{C}}_ n$ and let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the shift functor (given on objects by the construction $\{ C_ n \} _{n \geq 0} \mapsto \{ F_{n+1}( C_{n+1}) \} _{n \geq 0})$, so that we have a pullback diagram of simplicial sets
4.45
\begin{equation} \begin{gathered}\label{equation:sequential-limit-as-pullback} \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}}. } \end{gathered} \end{equation}
If each of the functors $F_{n}$ is an isofibration, then (4.45) is a categorical pullback square of $\infty $-categories.
Proof.
We define a sequence of $\infty $-categories $\{ \operatorname{\mathcal{D}}_ n \} _{n \geq 0}$ and isofibrations $U_ n: \operatorname{\mathcal{D}}_ n \rightarrow \operatorname{\mathcal{C}}_ n$ as follows:
For $n = 0$, we define $\operatorname{\mathcal{D}}_ n$ to be $\operatorname{\mathcal{C}}_ n$ and take $U_ n$ to be the identity functor.
For $n > 0$, we define $\operatorname{\mathcal{D}}_{n}$ to be the homotopy fiber product $\operatorname{\mathcal{C}}_{n} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}_{n-1} } \operatorname{\mathcal{D}}_{n-1}$ and take $U_ n$ to be the map given by projection onto the first factor (which is an isofibration by virtue of Remark 4.5.3.2).
Each of the isofibrations $U_{n}$ admits a section $T_ n: \operatorname{\mathcal{C}}_{n} \rightarrow \operatorname{\mathcal{D}}_{n}$, given for $n > 0$ by the composition
\[ \operatorname{\mathcal{C}}_{n} \simeq \operatorname{\mathcal{C}}_{n} \times _{\operatorname{\mathcal{C}}_{n-1}} \operatorname{\mathcal{C}}_{n-1} \hookrightarrow \operatorname{\mathcal{C}}_{n} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}_{n-1} } \operatorname{\mathcal{C}}_{n-1} \xrightarrow { \operatorname{id}\times T_{n-1} } \operatorname{\mathcal{C}}_{n} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}_{n-1}} \operatorname{\mathcal{D}}_{n-1} = \operatorname{\mathcal{D}}_{n}. \]
Using induction on $n$ (together with Corollaries 4.5.3.29 and 4.5.3.21), we see that each of the functors $T_{n}$ is an equivalence of $\infty $-categories. Setting $\operatorname{\mathcal{C}}_{-1} = \Delta ^0$, we note that (4.45) can be identified with the outer rectangle of a diagram of pullback squares
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim \operatorname{\mathcal{C}}_{n} \ar [r]^-{ \varprojlim T_ n} \ar [d] & \varprojlim \operatorname{\mathcal{D}}_ n \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{(\operatorname{id},\operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r] & \prod _{n \geq 0} \operatorname{\mathcal{C}}_{n} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}_{n-1}} \operatorname{\mathcal{C}}_{n-1} \ar [r] & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \]
where the horizontal map on the bottom left is given by the product of the inclusion maps
\[ \operatorname{\mathcal{C}}_{n} \simeq \operatorname{\mathcal{C}}_{n} \times _{ \operatorname{\mathcal{C}}_{n-1} } \operatorname{\mathcal{C}}_{n-1} \hookrightarrow \prod _{n \geq 0} \operatorname{\mathcal{C}}_{n} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}_{n-1}} \operatorname{\mathcal{C}}_{n-1} \]
(which is an equivalence of $\infty $-categories by virtue of Corollary 4.5.3.29) and the horizontal map on the bottom left is given by the product of the forgetful functors
\[ \operatorname{\mathcal{C}}_{n} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}_{n-1}} \operatorname{\mathcal{C}}_{n-1} \rightarrow \operatorname{\mathcal{C}}_{n} \times \operatorname{\mathcal{C}}_{n-1} \]
(which is an isofibration by virtue of Remark 4.5.3.2). Invoking Corollary 4.5.3.28, we see that the right side of the diagram is a categorical pullback square. It follows that the outer rectangle is a categorical pullback square if and only if the horizontal map on the upper left is an equivalence of $\infty $-categories (Proposition 4.5.3.20). This follows from Example 4.5.7.18, since the map arises from a natural transformation of towers
\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r]^-{F_2} & \operatorname{\mathcal{C}}_2 \ar [r]^-{F_1} \ar [d]^{ T_2} & \operatorname{\mathcal{C}}_1 \ar [r]^-{ F_0 } \ar [d]^{ T_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{ T_0 } \\ \cdots \ar [r] & \operatorname{\mathcal{D}}_2 \ar [r] & \operatorname{\mathcal{D}}_1 \ar [r] & \operatorname{\mathcal{D}}_0 } \]
where the horizontal maps are isofibrations (see Remark 4.5.3.2) and the vertical maps are equivalences of $\infty $-categories.
$\square$