Corollary 9.2.3.6. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$ if and only if the following conditions are satisfied:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ admits small filtered colimits.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
For each object $X \in \operatorname{\mathcal{C}}$, the image $f(X)$ is a compact object of $\operatorname{\mathcal{D}}$.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated under small filtered colimits by the image of $f$.
Moreover, if $f$ satisfies conditions $(0)$, $(1)$, and $(2)$, then its $\operatorname{Ind}$-extension $F: \operatorname{Ind}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ is fully faithful.