Definition 4.8.0.3 (Preliminary). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration $\infty $-categories and let $m \geq 0$ be an integer. We say that $F$ is $m$-full if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^ m \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]
admits a solution.