$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 8.4.5.9. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and define $\widehat{\operatorname{\mathcal{C}}}$ as in Construction 8.4.5.5. Let $\operatorname{\mathcal{D}}$ be a $\mathbb {K}$-cocomplete $\infty $-category and let $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from the essential image of the Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$.
- $(2)$
The functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
Proof.
Let $F_0$ denote the restriction of $F$ to the essential image of $h_{\bullet }$. Applying Lemma 8.4.5.8, we deduce that $F_0$ admits a left Kan extension $F': \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed colimits for each $K \in \mathbb {K}$. Invoking the universal property of Kan extensions (Corollary 7.3.6.9), we see that there is an essentially unique natural transformation $\alpha : F' \rightarrow F$ which restricts to the identity transformation from $F_0$ to itself. We can then reformulate condition $(1)$ as follows:
- $(1')$
The natural transformation $\alpha $ is an isomorphism. That is, for each object $X \in \widehat{\operatorname{\mathcal{C}}}$, the induced map $\alpha _{X}: F'(X) \rightarrow F(X)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.
The implication $(1') \Rightarrow (2)$ follows from the fact that $F'$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. To prove the converse, let $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ denote the full subcategory spanned by those objects $X$ for which $\alpha _{X}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. By construction, $\widehat{\operatorname{\mathcal{C}}}'$ contains all representable functors $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. If condition $(2)$ is satisfied, then $\widehat{\operatorname{\mathcal{C}}}'$ is closed under the formation of $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore coincides with $\widehat{\operatorname{\mathcal{C}}}$.
$\square$