Remark 4.8.6.33. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $n$ be an integer. Then $F$ is an $n$-categorical inner fibration if and only if, for every pullback diagram of simplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{ F } \\ \Delta ^{m} \ar [r] & \operatorname{\mathcal{D}}, } \]
the projection map $F'$ is an $n$-categorical inner fibration. For $n \geq 0$, this is equivalent to the requirement that $\operatorname{\mathcal{C}}'$ is an $(n,1)$-category (Proposition 4.8.6.32).