# Kerodon

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

Corollary 7.3.2.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\beta : F_0 \rightarrow F|_{ \operatorname{\mathcal{C}}^0 }$ be a natural transformation of functors from $\operatorname{\mathcal{C}}^{0}$ to $\operatorname{\mathcal{D}}$. Then $\beta$ exhibits $F$ as a left Kan extension of $F_0$ along the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ (in the sense of Variant 7.3.1.5) if and only if the following pair of conditions is satisfied:

$(1)$

The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1).

$(2)$

The natural transformation $\beta$ is an isomorphism in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$.

Proof. By virtue of Corollary 7.3.1.16, we may assume that condition $(2)$ is satisfied. Using Remark 7.3.1.10, we can reduce further to the special case where $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$ and $\beta$ is the identity transformation, in which case the desired result is a restatement of Proposition 7.3.2.5. $\square$