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Proposition Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 1$ be an integer. Then $F$ is $n$-full if and only if every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]

admits a solution. If $F$ is an isofibration, then this is also true in the case $n = 0$.

Proof. The case $n=0$ reduces to the observation that an isofibration is essentially surjective if and only if it is surjective on objects. The case $n = 1$ is a reformulation of Variant We may therefore assume without loss of generality that $n \geq 2$. Using Proposition, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, the functor $F$ is $n$-full if and only if, for every morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u )$ consists of a single element (Corollary The desired result now follows from Corollary $\square$