# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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.