Variant 11.10.1.11. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cartesian fibrations of simplicial sets. We will say that a morphism of simplicial sets $f: X \rightarrow X'$ is an equivalence of cartesian fibrations over $S$ if it satisfies the following conditions:
The morphism $f$ carries $q$-cartesian edges of $X$ to $q'$-cartesian edges of $X'$ and satisfies $q = q' \circ f$; that is, $f$ is a morphism in the category $\operatorname{Cart}(S)$.
The homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Cart}(S)}$.
We say that the cartesian fibrations $q: X \rightarrow S$ and $q': X' \rightarrow S$ are equivalent if there exists a morphism $f: X \rightarrow X'$ which is an equivalence of cartesian fibrations over $S$: that is, if $X$ and $X'$ are isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{Cart}(S)}$.