Proposition 4.2.4.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then:

- $(1)$
The morphism $q$ is a left fibration if and only if, 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 left anodyne, there exists a dotted arrow rendering the diagram commutative.

- $(2)$
The morphism $q$ is a right fibration if and only if, 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 right anodyne, there exists a dotted arrow rendering the diagram commutative.