Kerodon

Example 11.10.1.5. Let $S$ be a Kan complex. Then a morphism of simplicial sets $q: X \rightarrow S$ is a (co)cartesian fibration if and only if it is an isofibration (Corollary 5.1.4.11). If this condition is satisfied, then $X$ is an $\infty $-category and a morphism of $X$ is $q$-(co)cartesian if and only if it is an isomorphism (Corollary 5.1.1.11). It follows that, if $q': X' \rightarrow S$ is another cartesian fibration, then every morphism $f: X \rightarrow X'$ in $(\operatorname{Set_{\Delta }})_{/S}$ carries $q$-(co)cartesian edges of $X$ to $q'$-(co)cartesian edges of $X'$ (Remark 1.5.1.6). In particular, $\operatorname{Cart}(S)$ coincides with $\operatorname{CCart}(S)$ as a simplicial subcategory of $(\operatorname{Set_{\Delta }})_{/S}$.