$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 8.5.8.6. Let $n \geq 3$ be an integer, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a partial idempotent $F_{< n}: \operatorname{N}_{\leq n-1}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $F_{< n}$ extends to an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.
- $(2)$
The morphism $F_{< n}$ extends to a partial idempotent $F_{\leq n}: \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.
Proof.
The implication $(1) \Rightarrow (2)$ is immediate. To prove the converse, suppose that $F_{\leq n}: \operatorname{N}_{\leq n}(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ is an extension of $F_{< n}$. Let $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$, $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, and $V: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ be as in Theorem 8.5.8.4. Since $\iota $ is inner anodyne, we can choose a functor $\overline{F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F} \circ \iota = F_{\leq n}$. Then $F = \overline{F} \circ V$ is a functor from $\operatorname{N}_{\bullet }( \operatorname{Idem})$ to $\operatorname{\mathcal{C}}$, and Remark 8.5.8.5 shows that $F$ coincides with $F_{< n}$ on the $(n-1)$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$.
$\square$