Kerodon

Remark 7.3.3.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which is isomorphic to $F$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Proposition 7.1.5.13). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.