Definition 9.2.1.4. Let $\kappa $ be a small regular cardinal. We say that a functor of $\infty $-categories $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(a)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits small $\kappa $-filtered colimits.
- $(b)$
Let $\operatorname{\mathcal{D}}$ be any $\infty $-category which admits small $\kappa $-filtered colimits. Then precomposition with $H$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\kappa -\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Here $\operatorname{Fun}^{\kappa -\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ denotes the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by the $\kappa $-finitary functors (Definition 9.1.9.3).