Remark 5.1.3.5. Suppose we are given a pullback diagram of simplicial sets

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

Let $e'$ be an edge of the simplicial set $X'$, having image $e = f(e')$ in $X$. Then $e$ is locally $q$-cartesian if and only if $e'$ is locally $q'$-cartesian. Similarly, $e$ is locally $q$-cocartesian if and only if $e'$ is locally $q'$-cocartesian.