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Lemma 9.4.1.23. Let $\widehat{\operatorname{\mathcal{D}}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ be a full subcategory. Suppose that the inclusion functor $\operatorname{\mathcal{D}}\hookrightarrow \widehat{\operatorname{\mathcal{D}}}$ exhibits $\widehat{\operatorname{\mathcal{D}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{D}}$, for some collection of simplicial sets $\mathbb {K}$. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocomplete inner fibration of $\infty $-categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $F: \widehat{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{E}}_{C}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
- $(2)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{D}}$.
- $(3)$
When regarded as a functor from $\widehat{\operatorname{\mathcal{D}}}$ to $\operatorname{\mathcal{E}}$, the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{D}}$.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Lemma 8.4.5.9, and the implication $(3) \Rightarrow (2)$ is a special case of Corollary 7.3.3.23. We will complete the proof by showing that $(1)$ implies $(3)$. By virtue of Remark 7.3.3.24, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $C$ is the initial vertex; in this case, we wish to show that $F$ is left Kan extended from $\operatorname{\mathcal{D}}$ (when regarded as a functor from $\widehat{\operatorname{\mathcal{D}}}$ to $\operatorname{\mathcal{E}}$). Fix an uncountable regular cardinal $\kappa $ such that every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. Let $\lambda $ be a cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{E}}$ is locally $\lambda $-small. By virtue of Proposition 7.4.1.18, it will suffice to show that for every object $E \in \operatorname{\mathcal{E}}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{D}}} \rightarrow (\operatorname{\mathcal{S}}^{< \lambda })^{\operatorname{op}} \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{E}}}( F(D), E ) \]
is left Kan extended from $\operatorname{\mathcal{D}}$. By virtue of Lemma 8.4.5.9, this is equivalent to the requirement that $\mathscr {F}$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. This follows from $(1)$ together with our assumption that $U$ is $\mathbb {K}$-cocomplete.
$\square$