$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 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 $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits. Then $F_0$ admits a left Kan extension along $\delta $.

Proof. By virtue of Proposition, 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 colimit in the $\infty $-category $\operatorname{\mathcal{D}}$. Note that the projection map $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is a right fibration of simplicial sets (Proposition, whose fiber over each vertex $x \in K$ can be identified with the Kan complex $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( \delta (x), C)$. Invoking Proposition, we see that $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( \delta (x), C)$ is homotopy equivalent to the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \delta (x), C)$, 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 implies that $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is essentially $\kappa $-small. The desired result now follows from our assumption that $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits (Remark $\square$