Kerodon

Remark 7.3.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ (in the sense of Definition 7.3.2.1) if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ (in the sense of Definition 7.3.3.1), where $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ is the projection map. Similarly, $F$ is right Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$. See Example 7.1.5.3.