$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.6.4.22 (Categorial Pullbacks and Slices). Suppose we are given a categorical pullback diagram of $\infty $-categories
4.55
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square-slice} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r]^-{G} \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G'} & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}
For any diagram $q: K \rightarrow \operatorname{\mathcal{C}}'$, the diagram of slice $\infty $-categories
4.56
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square-slice2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'_{/q} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ / (G \circ q)} \ar [d] \\ \operatorname{\mathcal{D}}'_{ / (F' \circ q)} \ar [r] & \operatorname{\mathcal{D}}_{ / (F \circ G \circ q)}. } \end{gathered} \end{equation}
is also a categorical pullback square.
Proof.
Using Corollary 4.5.3.24, we can factor $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {E} \overline{\operatorname{\mathcal{C}}} \xrightarrow { \overline{F} } \operatorname{\mathcal{D}}$, where $E$ is an equivalence of $\infty $-categories and $\overline{F}$ is an isofibration. Using Proposition 4.5.3.20 and Corollary 4.6.4.21, we can replace $\operatorname{\mathcal{C}}$ by $\overline{\operatorname{\mathcal{C}}}$ and thereby reduce to proving Corollary 4.6.4.22 in the special case where $F$ is an isofibration. In this case, our assumption that (4.55) is a categorical pullback square guarantees that the induced map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$ is an equivalence of $\infty $-categories (Proposition 4.5.3.27). Using Proposition 4.5.3.20 and Corollary 4.6.4.21 again, we can replace $\operatorname{\mathcal{C}}'$ by the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$ and thereby reduce to the situation where (4.55) is a pullback diagram of simplicial sets. In this case, (4.56) is also a pullback diagram of simplicial sets in which the vertical maps are isofibrations (Corollary 4.4.3.21). In this case, the desired result follows from Corollary 4.5.3.28.
$\square$