$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.1.7.7. Suppose we are given a categorical pullback square of $\infty $-categories
7.5
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-relative-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{F} \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}'$ be a diagram. If $F \circ \overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$, then $\overline{q}$ is a $U'$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}'$.
Proof.
Using Corollary 4.5.3.24, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {E} \overline{\operatorname{\mathcal{D}}} \xrightarrow { V} \operatorname{\mathcal{D}}$, where $V$ is an isofibration and $E$ is an equivalence of $\infty $-categories. Applying Remark 7.1.6.6, we conclude that $(E \circ F \circ \overline{f}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $V$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. We may therefore replace $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{D}}$ and thereby reduce to proving Corollary 7.1.7.7 in the situation where $U$ is an isofibration. In this case, our assumption that (7.5) is a categorical pullback square guarantees that the induced map $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an equivalence of $\infty $-categories. Using Remark 7.1.6.6 again, we can replace $\operatorname{\mathcal{E}}'$ by $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and thereby reduce to proving Corollary 7.1.7.7 in the situation where (7.5) is a pullback square (and the vertical maps are isofibrations). In this case, the desired result follows from Proposition 7.1.7.6.
$\square$