Variant 4.8.6.19. Let $n \geq -1$ be an integer and suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix { \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \operatorname{\mathcal{E}}, & } \]
where the vertical maps are inner fibrations. Then $F$ is essentially $n$-categorical if and only if, for every morphism $u$ of $\operatorname{\mathcal{E}}$, the induced functor
\[ F_{u}: \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]
is essentially $n$-categorical. In particular, $F$ is fully faithful if and only if each $F_ u$ is fully faithful.