Remark 5.1.3.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. Suppose that $S$ is isomorphic to a left cone $K^{\triangleleft }$ and that $q$ carries the vertex $x \in X$ to the cone point of $K^{\triangleleft }$. Then $e$ is $q$-cartesian if and only if it is locally $q$-cartesian. Similarly, if $S$ is isomorphic to a right cone $L^{\triangleright }$ and $q$ carries the vertex $y \in X$ to the cone point of $L^{\triangleright }$, then $e$ is $q$-cocartesian if and only if it is locally $q$-cocartesian.

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