Example 9.2.1.10 (Idempotent Completion). Let $\kappa $ be a regular cardinal and let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. Then:
If $\kappa $ is uncountable, then $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\kappa }$-completion of $\operatorname{\mathcal{C}}$ if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$.
If $\kappa = \aleph _0$, then $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\kappa }$-completion of $\operatorname{\mathcal{C}}$ if and only if it is an equivalence of $\infty $-categories.
See Proposition 9.1.9.17.