Example 4.8.8.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} \xrightarrow {G} \Delta ^0, \]
where $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is the homotopy $n$-category constructed in ยง4.8.4. This factorization satisfies the requirements of Theorem 4.8.8.3: the functor $G$ is essentially $n$-categorical because $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is an $(n,1)$-category (Example 4.8.6.4), and the functor $F'$ is categorically $(n+1)$-connective by Example 4.8.5.12.