Proposition 5.1.3.11 (01U8)

Proposition 5.1.3.11. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, and let $e: y \rightarrow z$ be an edge of the simplicial set $X$ having image $\overline{e}: s \rightarrow t$ in $S$. Then $e$ is locally $q$-cartesian if and only if, for every object $x$ of the $\infty $-category $X_{s}$, the composition map

\[ \operatorname{Hom}_{X_ s}( x,y ) \xrightarrow { [e] \circ } \operatorname{Hom}_{X}(x,z)_{\overline{e} } \]

of Notation 5.1.3.10 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Proof. Without loss of generality, we can replace $q: X \rightarrow S$ by the projection map $X \times _{S} \Delta ^1 \rightarrow \Delta ^1$ and thereby reduce to the case where $S = \Delta ^1$ and $\overline{e}$ is the unique nondegenerate edge of $\Delta ^1$. In this case, the edge $e$ is locally $q$-cartesian if and only if it is $q$-cartesian, and the desired result is a special case of Corollary 5.1.2.3. $\square$