Corollary 5.1.5.12. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. The following conditions are equivalent:
The morphism $q$ is a right fibration.
For every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex.
Every edge of $X$ is locally $q$-cartesian.
Every edge of $X$ is $q$-cartesian.
Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 4.4.2.3. To show that $(2) \Rightarrow (3)$, we may 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 5.1.4.15. The implication $(3) \Rightarrow (4)$ follows from Corollary 5.1.5.10. If condition $(4)$ is satisfied, then $q$ is a cartesian fibration (Corollary 5.1.5.10), so that $(1)$ follows from Proposition 5.1.4.15. $\square$