Definition 7.3.2.1. 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. Fix an object $C \in \operatorname{\mathcal{C}}$. We will say that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$ if the composite map
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. Here $\operatorname{\mathcal{C}}^{0}_{/C}$ denotes the fiber product $\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ (Notation 7.3.1.1), and $c$ is the slice contraction morphism of Construction 4.3.5.12. Similarly, we say that $F$ is right Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$ if the composite map
is a limit diagram in $\operatorname{\mathcal{D}}$. We say that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ if it is left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$. We say that $F$ is right Kan extended from $\operatorname{\mathcal{C}}^0$ if it is right Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$.