Kerodon

Definition 7.3.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$. We will say that $F$ is a left Kan extension of $F_0$ if $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{\operatorname{\mathcal{C}}^{0}} = F_0$. We will say that $F$ is a right Kan extension of $F_0$ if $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{\operatorname{\mathcal{C}}^{0}} = F_0$.