Proposition 7.6.6.13. Let $\kappa $ be an uncountable regular cardinal and let $K$ be a simplicial set which is essentially $\kappa $-small. Suppose we are given a pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, together with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$. Suppose that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and that $\operatorname{\mathcal{D}}$ is $\kappa $-complete. Then $F_0$ admits a right Kan extension along $\delta $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 7.3.5.1, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the composite map
\[ K \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/} \rightarrow K \xrightarrow { F_0 } \operatorname{\mathcal{D}} \]
admits a limit in the $\infty $-category $\operatorname{\mathcal{D}}$. Note that the projection map $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is a left fibration of simplicial sets (Proposition 4.3.6.1), whose fiber over each vertex $x \in K$ can be identified with the Kan complex $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( C,\delta (x) )$. Invoking Proposition 4.6.5.10, we see that $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( C,\delta (x) $ is homotopy equivalent to the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \delta (x))$, and is therefore essentially $\kappa $-small (by virtue of our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small). Since $K$ is essentially $\kappa $-small, Corollary 5.6.7.7 implies that the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is essentially $\kappa $-small. The desired result now follows from our assumption that $\operatorname{\mathcal{D}}$ is $\kappa $-complete (Remark 7.6.6.6). $\square$