Kerodon

Example 9.2.1.6. Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. Then $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.2.1.1) if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\aleph _0}$-completion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.2.1.4). Stated more informally, we have an equivalence of $\infty $-categories $\operatorname{Ind}(\operatorname{\mathcal{C}}) \simeq \operatorname{Ind}_{\aleph _0}(\operatorname{\mathcal{C}})$.