Lemma 8.4.5.8 (04BR)

Lemma 8.4.5.8. 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 $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete. Then there exists a functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and an isomorphism $f \rightarrow F|_{\operatorname{\mathcal{C}}}$, where the functor $F$ is left Kan extended from the essential image of the Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$. Moreover, the functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.

Proof. Fix an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and each $K \in \mathbb {K}$ is essentially $\kappa $-small. By virtue of Corollary 8.3.3.17, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a replete full subcategory of a $\kappa $-cocomplete $\infty $-category $\operatorname{\mathcal{D}}'$, and that the inclusion map $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{D}}'$ preserves all $\kappa $-small colimits which exist in $\operatorname{\mathcal{D}}$. By virtue of Theorem 8.4.3.3, we can also assume that $f$ factors as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {F'} \operatorname{\mathcal{D}}', \]

where $F'$ preserves $\kappa $-small colimits. The full subcategory $F'^{-1}( \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ contains all representable functors and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore contains $\widehat{\operatorname{\mathcal{C}}}$. It follows that $F'$ restricts to a functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed colimits for each $K \in \mathbb {K}$. Theorem 8.4.3.6 implies that $F'$ is left Kan extended from the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$, so the functor $F = F'|_{ \widehat{\operatorname{\mathcal{C}}} }$ has the same property. $\square$