$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.4.1.25. Let $\mathbb {K}$ be a collection of simplicial sets. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \widehat{\operatorname{\mathcal{E}}} \ar [rr]^-{F} \ar [dr]_{ \widehat{U} } & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}& } \]
and let $\operatorname{\mathcal{E}}\subseteq \widehat{\operatorname{\mathcal{E}}}$ be a full subcategory. Assume that $U = \widehat{U}|_{\operatorname{\mathcal{E}}}$ is an isofibration, that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \widehat{\operatorname{\mathcal{E}}}$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, and that $U'$ is a $\mathbb {K}$-cocomplete inner fibration. The following conditions are equivalent:
- $(1)$
The functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$.
- $(2)$
For each object $C \in \operatorname{\mathcal{C}}$ and each $K \in \mathbb {K}$, the functor $F_{C}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ preserves $K$-indexed colimits.
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$, and let $X$ be an object of $\widehat{\operatorname{\mathcal{E}}}$ satisfying $\widehat{U}(X) = C$. Combining Lemma 9.4.1.24 with Corollary 7.2.2.2, we see that the following conditions are equivalent:
- $(1_ X)$
The functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$ at the object $X \in \widehat{\operatorname{\mathcal{E}}}$.
- $(2_ X)$
The functor $F_ C$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}_{C}$ at the object $X \in \widehat{\operatorname{\mathcal{E}}}_{C}$.
Allowing the object $X$ to vary (with $C$ fixed) and applying Lemma 9.4.1.23, we see that the following are equivalent:
- $(1_ C)$
For each object $X \in \widehat{\operatorname{\mathcal{E}}}$ satisfying $\widehat{U}(X) = C$, the functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$ at the object $X$.
- $(2_ C)$
For each $K \in \mathbb {K}$, the functor $F_{C}$ preserves $K$-indexed colimits.
The equivalence $(1) \Leftrightarrow (2)$ now follows by allowing the object $C \in \operatorname{\mathcal{C}}$ to vary.
$\square$