$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Suppose we are given a pullback diagram of simplicial sets

\begin{equation} \begin{gathered}\label{equation:pullback-cocartesian-fibration} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{F} } \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{C}}', } \end{gathered} \end{equation}

where $U'$ is a cocartesian fibration. If $F$ is a categorical equivalence of simplicial sets, then $\widetilde{F}$ is also a categorical equivalence of simplicial sets.

Proof. Using Corollary, we can extend (5.53) to a diagram of pullback squares

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{F} } \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}'' \ar [d]^{U''} \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [r]^-{G} & \operatorname{\mathcal{C}}'' } \]

where $U''$ is a cocartesian fibration, $G$ is inner anodyne, and $\operatorname{\mathcal{C}}''$ is an $\infty $-category. Since $G$ and $G \circ F$ are categorical equivalences, it follows from Proposition that $\widetilde{G}$ and $\widetilde{G} \circ \widetilde{F}$ are categorical equivalences. Applying the two-out-of-three property (Remark, we conclude that $\widetilde{F}$ is also a categorical equivalence. $\square$