Corollary 5.1.1.16. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. The following conditions are equivalent:
- $(1)$
The edge $e$ is $q$-cartesian.
- $(2)$
Let $f: B \rightarrow X$ be a morphism of simplicial sets, let $A$ be a simplicial subset of $B$, and let $Y$ denote the fiber product $X_{f|_{A} / } \times _{ S_{ (q \circ f|_{A})/} } S_{ (q \circ f)/ }$, so that the restriction map $X_{f/} \rightarrow X$ factors as a composition $X_{f/} \xrightarrow {\theta } Y \xrightarrow {\rho } X$. Then every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \ar [r] \ar [d] & X_{f/} \ar [d]^-{\theta } \\ \Delta ^1 \ar [r]^-{e'} \ar@ {-->}[ur] & Y } \]admits a solution, provided that $\rho (e') = e$.