Remark 9.2.3.15 ($\operatorname{Ind}$-Completions of Ordinary Categories). Let $\operatorname{\mathcal{C}}$ be an ordinary category. It follows from Corollary 9.2.3.14 that there is a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ for which the induced map $\operatorname{N}_{\bullet }(H): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }( \widehat{\operatorname{\mathcal{C}}} )$ exhibits $\operatorname{N}_{\bullet }( \widehat{\operatorname{\mathcal{C}}} )$ as an $\operatorname{Ind}$-completion of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. We then have the following:
- $(a)$
The category $\widehat{\operatorname{\mathcal{C}}}$ admits small filtered colimits.
- $(b)$
For every category $\operatorname{\mathcal{D}}$ which admits small filtered colimits, composition with $H$ induces an equivalence
\[ \operatorname{Fun}^{\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}), \]where $\operatorname{Fun}^{\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ is the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve small filtered colimits.
These properties determine the category $\widehat{\operatorname{\mathcal{C}}}$ uniquely up to equivalence. To emphasize the uniqueness, we denote $\widehat{\operatorname{\mathcal{C}}}$ by $\operatorname{Ind}( \operatorname{\mathcal{C}})$ and refer to it as the $\operatorname{Ind}$-completion of the category $\operatorname{\mathcal{C}}$. Corollary 9.2.3.14 then supplies an equivalence of $\infty $-categories
In other words, the category $\operatorname{Ind}(\operatorname{\mathcal{C}})$ automatically satisfies a stronger version of condition $(b)$, where we allow $\operatorname{\mathcal{D}}$ to be an $\infty $-category.