Proposition 7.3.3.7. 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 and let $C \in \operatorname{\mathcal{C}}$ be an object which is isomorphic to an object of $\operatorname{\mathcal{C}}^{0}$. Then $F$ is both $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ and $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$; the analogous statement for the right Kan extension condition follows by a similar argument. Let $c: (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the slice contraction morphism; we wish to show that the composition $(F \circ c): (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram. Choose an object $C' \in \operatorname{\mathcal{C}}^{0}$ and an isomorphism $u: C' \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $u$ is an isomorphism guarantees that it is final when viewed as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.7.22), and therefore also when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/C}^{0}$. The desired result now follows from Corollary 7.2.3.6, since $F(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. $\square$