Proposition 8.4.6.6. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially small. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.4.0.1) if and only if the following conditions are satisfied:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is locally small and cocomplete.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
For each object $C \in \operatorname{\mathcal{C}}$, the image $f(C) \in \operatorname{\mathcal{D}}$ is atomic.
- $(3)$
The collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$.
Proof of Proposition 8.4.6.6.
Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category, let $\operatorname{\mathcal{D}}$ be a cocomplete $\infty $-category, and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. By virtue of Theorem 8.4.0.3, the functor $f$ admits an essentially unique factorization as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {F} \operatorname{\mathcal{D}}, \]
where $h_{\bullet }$ is the covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ and the functor $F$ preserves small colimits. Moreover, $f$ exhibits $\operatorname{\mathcal{D}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ if and only if the functor $F$ is an equivalence of $\infty $-categories. If this condition is satisfied, then $\operatorname{\mathcal{D}}$ is locally small (Corollary 4.7.8.9), the functor $f$ is fully faithful (Theorem 8.3.3.13), the essential image of $f$ consists of atomic objects of $\operatorname{\mathcal{D}}$ (Example 8.4.6.2). We may therefore assume without loss of generality that $f$ satisfies conditions $(0)$, $(1)$, and $(2)$ of Proposition 8.4.6.6, so that $F$ is fully faithful (Lemma 8.4.6.8). Using Proposition 8.4.4.1, we see that the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, given on objects by the formula $G(D)(C) = \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D)$. By virtue of Corollary 6.2.2.21, the functor $F$ is an equivalence if and only if the functor $G$ is conservative: that is, if and only if the collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense.
$\square$