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

Corollary Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then $U$ is exponentiable (Definition In particular, for any pullback diagram of simplicial sets

\begin{equation} \begin{gathered}\label{equation:pullback-cocartesian-fibration} \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]^-{\overline{F}} & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

if $\overline{F}$ is a categorical equivalence, then $F$ is also a categorical equivalence.

Proof. By virtue of Corollary and Remark, we may assume that $U$ is a cocartesian fibration of $\infty $-categories, in which case the desired result follows from Proposition $\square$