Kerodon

Warning 8.5.8.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F_{\leq n}: \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be a partial idempotent in $\operatorname{\mathcal{C}}$. Corollary 8.5.8.6 asserts that, if $n \geq 3$, then we can choose an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ such that $F$ and $F_{\leq n}$ coincide on the $(n-1)$-skeleton $\operatorname{N}_{\bullet }( \operatorname{Idem})$. Beware that we generally cannot arrange that $F|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) }$ coincides with $F_{\leq n}$. For example, this always fails in the (universal) case $F_{\leq n}$ is the inner anodyne morphism $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$ of Theorem 8.5.8.4 (see Remark 8.5.8.5).