Example 4.8.5.12. Let $n \geq -2$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{D}}$ (see Definition 4.8.2.9). Then $F$ is $m$-full for $m \leq n+2$. In particular, for any $\infty $-category $\operatorname{\mathcal{C}}$, the canonical maps
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \quad \quad \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n+1}^{\circ }(\operatorname{\mathcal{C}}) \quad \quad \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n+1}}\mathit{(\operatorname{\mathcal{C}})} \]
are $m$-full for $m \leq n+2$.