Remark 9.2.1.21 (0645)

Remark 9.2.1.21. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Choose a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\mu $-small, and let

\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \]

be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. By virtue of Example 8.4.4.5, left Kan extension along $F$ determines a $\mu $-cocontinuous functor $F_{!}: \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{ < \mu } )$ for which the diagram

9.3

\begin{equation} \begin{gathered}\label{equation:Ind-functor-via-LKE} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{F} \ar [r]^{ h_{\bullet }^{\operatorname{\mathcal{C}}} } & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \ar [d]^{F_{!}} \\ \operatorname{\mathcal{D}}\ar [r]^{ h_{\bullet }^{\operatorname{\mathcal{D}}} } & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) } \end{gathered} \end{equation}

commutes up to (canonical) isomorphism. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ with the smallest full subcategories of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ and $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ which are closed under $\lambda $-small $\kappa $-filtered colimits and which contain the essential images of $h_{\bullet }^{\operatorname{\mathcal{C}}}$ and $h_{\bullet }^{\operatorname{\mathcal{D}}}$, respectively. It follows from the commutativity of the diagram (9.3) that $F_{!}$ restricts to a functor $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, which we can identify with the functor $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ of Remark 9.2.1.20.