Corollary 9.10.1.15. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^-{f} \ar [dr]_{q} & & X' \ar [dl]^{q'} \\ & S, & } \]

where the morphisms $q$ and $q'$ are both cartesian fibrations and cocartesian fibrations. Then $f$ is an equivalence of cartesian fibrations over $S$ if and only if $f$ is an equivalence of cocartesian fibrations over $S$.