Proposition 4.1.3.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if it satisfies the following condition:

- $(\ast )$
For every square diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]where $i$ is inner anodyne, there exists a dotted arrow rendering the diagram commutative.