Definition 5.1.4.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets. We say that $q$ is a cartesian fibration if the following conditions are satisfied:
- $(1)$
The morphism $q$ is an inner fibration.
- $(2)$
For every edge $\overline{e}: s \rightarrow t$ of the simplicial set $S$ and every vertex $z \in X$ satisfying $q(z) = t$, there exists a $q$-cartesian edge $e: y \rightarrow z$ of $X$ satisfying $q(e) = \overline{e}$.
We say that $q$ is a cocartesian fibration if it satisfies condition $(1)$ together with the following dual version of $(2)$:
- $(2')$
For every edge $\overline{e}: s \rightarrow t$ of the simplicial set $S$ and every vertex $y \in X$ satisfying $q(y) = s$, there exists a $q$-cocartesian edge $e: y \rightarrow z$ of $X$ satisfying $q( e) = \overline{e}$.