Definition 9.2.1.1. Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. We say that $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(a)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits small filtered colimits.
- $(b)$
Let $\operatorname{\mathcal{D}}$ be any $\infty $-category which admits small filtered colimits. Then precomposition with $H$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.