$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Variant 8.4.6.9. Let $\mathbb {K}$ be a collection of simplicial sets and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.4.5.1) if and only if the following conditions are satisfied:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
Let $\kappa $ be an uncountable regular cardinal such that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small and each $K \in \mathbf{K}$ is essentially $\kappa $-small. Then, for each $C \in \operatorname{\mathcal{C}}$, the corepresentable functor
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D ) \]
preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated by the objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ under the formation of $K$-indexed colimits for $K \in \mathbb {K}$.
Proof.
Let $\kappa $ be as in $(2)$, and let $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the smallest replete full subcategory which contains all representable functors and is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$ (see Construction 8.4.5.5). By virtue of Proposition 8.4.5.7, the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Assume that $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-complete, so that $f$ factors (up to isomorphism) as a composition $\operatorname{\mathcal{C}}\xrightarrow {h_{\bullet }} \widehat{\operatorname{\mathcal{C}}} \xrightarrow {F} \operatorname{\mathcal{D}}$, where the functor $F$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. To complete the proof, it will suffice to show that if $f$ satisfies conditions $(1)$, $(2)$, and $(3)$, then the functor $F$ is an equivalence of $\infty $-categories (the reverse implication follows from Theorem 8.3.3.13 and Example 8.4.6.2). Applying Lemma 8.4.6.8, we see that the functor $F$ is fully faithful and therefore restricts to an equivalence of $\widehat{\operatorname{\mathcal{C}}}$ with a replete full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$. For each $K \in \mathbb {K}$, our assumption that $F$ preserves $K$-indexed colimits guarantees that the subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is closed under the formation of $K$-indexed colimits. Since $\operatorname{\mathcal{D}}_0$ contains the essential image of the functor $f$, the equality $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$ follows from assumption $(3)$.
$\square$