Kerodon

Remark 7.3.3.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor which is isomorphic to $U$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$). Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $V$-left Kan extended from $\operatorname{\mathcal{C}}^0$ (see Remark 7.1.6.7). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $V$-right Kan extended from $\operatorname{\mathcal{C}}^0$.