Corollary 11.10.1.17. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}. & } \]
If $U$ and $U'$ are cartesian fibrations, then $F$ is an equivalence of cartesian fibrations over $\operatorname{\mathcal{C}}$ if and only if it is an equivalence of $\infty $-categories. If $U$ and $U'$ are cocartesian fibrations, then $F$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ if and only if it is an equivalence of $\infty $-categories.