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Proposition 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.

Proof. The “if” direction is immediate from the definition, since the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is inner anodyne for $0 < i < n$. The reverse implication follows from Proposition $\square$