Corollary 4.1.3.4. Let $i: A \rightarrow B$ be a morphism of simplicial sets. Then $i$ is inner anodyne if and only if it satisfies the following condition:
For every square diagram of simplicial sets
where $q$ is an inner fibration, there exists a dotted arrow rendering the diagram commutative.
Proof. The “if” direction follows from Proposition 4.1.3.1. For the converse, suppose that condition $(\ast )$ is satisfied. Using Proposition 4.1.3.2, we can factor $i$ as a composition $A \xrightarrow {i'} Q \xrightarrow {q} B$, where $i'$ is inner anodyne and $q$ is an inner fibration. If the lifting problem
admits a solution, then the morphism $r$ exhibits $i$ as a retract of $i'$ (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). Since the collection of inner anodyne morphisms is closed under retracts, it follows that $i$ is inner anodyne. $\square$