Definition 4.8.6.24. Let $n$ be a positive integer. We say that a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an $n$-categorical inner fibration if it satisfies the following condition:
- $(\ast )$
For every pair of integers $0 < i < m$, every lifting problem
\[ \xymatrix@C =50pt@R=50pt{ \Lambda ^{m}_{i} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]admits a solution. Moreover, if $m > n$, then the solution is unique.