Remark 4.5.6.24. For small values of $n$, condition $(\ast _ n)$ of Proposition 4.5.6.23 can be stated more concretely:
For $n = 0$, it asserts that the simplicial set $\mathscr {F}( [0] )$ is an $\infty $-category.
For $n = 1$, it asserts that the face operators of $\mathscr {F}$ determine an isofibration of simplicial sets
\[ (d^{1}_{0}, d^{1}_{0} ): \mathscr {F}( [1] ) \rightarrow \mathscr {F}( [0] ) \times \mathscr {F}( [0] ). \]
If both of these conditions are satisfied, then the face operators $d^{1}_{0}, d^{1}_{1}: \mathscr {F}([1] ) \rightarrow \mathscr {F}( [0] )$ are isofibrations of $\infty $-categories.