Remark 5.1.4.6. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f} \ar [d]^-{q'} & X \ar [d]^-{q} \\ S' \ar [r] & S. } \]

If $q$ is a cartesian fibration, then $q'$ is also a cartesian fibration. Moreover, an edge $e'$ of $X'$ is $q'$-cartesian if and only if $e = f(e')$ is a $q$-cartesian edge of $X$ (this follows from Remarks 5.1.4.5 and 5.1.3.5). Similarly, if $q$ is a cocartesian fibration, then $q'$ is also a cocartesian fibration (and an edge $e'$ of $X'$ is $q'$-cocartesian if and only if $e = f(e')$ is a $q$-cocartesian edge of $X$).