Remark 8.5.6.19 (Cofinal Idempotents). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$, carrying the unique object of $\operatorname{Idem}$ to an object $X \in \operatorname{\mathcal{C}}$. Using Corollary 8.5.5.4, we can choose a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$. Then the idempotent $H \circ F$ splits: that is, it extends to a functor $\overline{F}: \operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \widehat{\operatorname{\mathcal{C}}}$ carrying the final object of $\operatorname{N}_{\bullet }(\operatorname{Ret})$ to an object $Y \in \widehat{\operatorname{\mathcal{C}}}$ which is a retract of $H(X)$. We then have a commutative diagram of $\infty $-categories
where the vertical maps are Morita equivalences. Using Corollaries 8.5.6.16, 8.5.6.17, and 7.2.1.9, we see that the following conditions are equivalent:
The idempotent $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ is a right cofinal functor.
The functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \widehat{\operatorname{\mathcal{C}}}$ is right cofinal.
The object $Y \in \widehat{\operatorname{\mathcal{C}}}$ is final.
Similarly, the idempotent $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ is left cofinal if and only if $Y$ is an initial object of $\widehat{\operatorname{\mathcal{C}}}$.