Kerodon

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

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. $\square$

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 $\widehat{\operatorname{\mathcal{C}}}$. $\square$

Proof of Proposition 8.4.5.7. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\widehat{\operatorname{\mathcal{C}}}$ be as in Construction 8.4.5.5. By construction, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete. To complete the proof, we must show that if $\operatorname{\mathcal{D}}$ is any $\mathbb {K}$-cocomplete $\infty $-category, then composition with the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ induces an equivalence of $\infty $-categories $\theta : \operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Let $\operatorname{\mathcal{C}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ be the essential image of $h_{\bullet }$, so that $\theta $ factors as a composition

where $\theta ''$ is an equivalence of $\infty $-categories (Theorem 8.3.3.13). Using Lemma 8.4.5.9, we see that $\operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ is the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which are left Kan extended from $\operatorname{\mathcal{C}}'$. It follows from Corollary 7.3.6.15 that $\theta '$ is a trivial Kan fibration onto a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}})$; in particular, it is fully faithful, so that $\theta $ is fully faithful. Lemma 8.4.5.8 implies that $\theta $ is essentially surjective, and therefore an equivalence of $\infty $-categories (Theorem 4.6.2.21). $\square$

Proof. It follows from Proposition 8.4.5.3 that the functor $h$ is fully faithful. We can therefore replace $\operatorname{\mathcal{C}}$ by its essential image and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ (and $h$ is the inclusion functor). In this case, condition $(1)$ is equivalent to the requirement that the functor $\operatorname{Tr}_{\operatorname{\mathcal{E}}/ \widehat{\operatorname{\mathcal{C}}} }$ is left Kan extended from $\operatorname{\mathcal{C}}$ (Lemma 8.4.5.9). The equivalence $(1) \Leftrightarrow (2)$ is now a special case of Corollary 7.4.3.15. $\square$