Definition 5.1.3.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e$ be an edge of $X$ having image $\overline{e} = q(e)$ in $S$. Form a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{e} \ar [r] \ar [d]^-{q'} & X \ar [d]^-{q} \\ \Delta ^1 \ar [r]^-{ \overline{e} } & S, } \]
so that $e$ lifts uniquely to an edge $\widetilde{e}$ of $X_{e}$ having nondegenerate image in $\Delta ^1$. We say that $e$ is locally $q$-cartesian if $\widetilde{e}$ is a $q'$-cartesian edge of the simplicial set $X_{e}$. We say that $e$ is locally $q$-cocartesian if $\widetilde{e}$ is a $q'$-cocartesian edge of the simplicial set $X_{e}$.