Kerodon

Example 7.3.3.6. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-left Kan extended from the empty subcategory $\emptyset \subseteq \operatorname{\mathcal{C}}$ if and only if it carries each object of $\operatorname{\mathcal{C}}$ to a $U$-initial object of $\operatorname{\mathcal{D}}$. Similarly, $F$ is $U$-right Kan extended from the empty subcategory if and only if it carries each object of $\operatorname{\mathcal{C}}$ to a $U$-final object of $\operatorname{\mathcal{D}}$.