Kerodon

Remark 5.1.5.7. Let $p: X \rightarrow Y$ and $q: Y \rightarrow Z$ be morphisms of simplicial sets. If $p$ is a cartesian fibration and $q$ is a locally cartesian fibration, then the composition $q \circ p$ is a locally cartesian fibration. Moreover, an edge $e$ of $X$ is locally $(q \circ p)$-cartesian if and only if it is $p$-cartesian and $p(e)$ is locally $q$-cartesian of $Y$. To prove this, we can assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration (Remark 5.1.5.6), so the desired result follows from Proposition 5.1.4.14.