Example 7.3.1.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $\alpha : F \rightarrow G$ be a morphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. The following conditions are equivalent:
- $(1)$
The natural transformation $\alpha $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
- $(2)$
The natural transformation $\alpha $ exhibits $F$ as a right Kan extension of $G$ along the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
The natural transformation $\alpha $ exhibits $G$ as a left Kan extension of $F$ along the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.
To prove the equivalence of $(1)$ and $(2)$, fix an object $C \in \operatorname{\mathcal{C}}$. Since the identity morphism $\operatorname{id}_{C}$ is an initial object of the $\infty $-category $\operatorname{\mathcal{C}}_{C/}$ (Proposition 4.6.7.22), the natural transformation $\alpha $ satisfies condition $(\ast _ C)$ of Definition 7.3.1.2 if and only if the induced map $\alpha _ C: F(C) \rightarrow G(C)$ is an isomorphism in $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.6). The equivalence $(1) \Leftrightarrow (2)$ now follows from the criterion of Theorem 4.4.4.4. The equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument.