Definition 7.3.3.1 (Relative Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. For each object $C \in \operatorname{\mathcal{C}}$, we will say that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$ if the composite map
is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. We say that $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$ if the composite map
is a $U$-limit diagram in $\operatorname{\mathcal{D}}$. Here $c$ and $c'$ denote the slice and coslice contraction morphisms of Construction 4.3.5.12. We say that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$. We say that $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$.