Remark 9.2.1.14 ($\operatorname{Ind}$-Extensions as Kan Extensions). In the situation of Definition 9.2.1.13, suppose that we are given functors $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\alpha : f \rightarrow F \circ H$. The following conditions are equivalent:
- $(1)$
The natural transformation $\alpha $ exhibits $F$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$, in the sense of Definition 9.2.1.13. That is, the functor $F$ is $(\kappa ,\lambda )$-finitary and $\alpha $ is an isomorphism.
- $(2)$
The natural transformation $\alpha $ exhibits $F$ as a left Kan extension of $f$ along $H$, in the sense of Variant 7.3.1.5.
See Corollary 8.4.5.11.