Corollary 4.8.5.28. Let $n > 0$ 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 $n$-full if and only if, for every morphism $u$ of $\operatorname{\mathcal{E}}$, the induced map
\[ F_{u}: \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]
is $n$-full.