Corollary 22.214.171.124. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, let $z$ be a vertex of $X$, and let $e: s \rightarrow q(z)$ be an edge of $S$. Suppose that there exists a $q$-cartesian edge $g: y \rightarrow z$ of $X$ satisfying $q(g) = e$. Then any locally $q$-cartesian edge $h: x \rightarrow z$ satisfying $q(h) = e$ is $q$-cartesian.
Proof. By virtue of Remark 126.96.36.199, we may assume without loss of generality that $S$ is an $\infty $-category (or even a simplex). Applying Remark 188.8.131.52, we deduce that there is a $2$-simplex of $X$ as depicted in the diagram