Warning 7.3.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion of a full subcategory, and let $F_0: \operatorname{\mathcal{C}}^0 \rightarrow \operatorname{\mathcal{D}}$. be a functor. We have given two definitions for the notion of Kan extension:
- $(a)$
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a left Kan extension of $F_0$ if it is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{\operatorname{\mathcal{C}}^{0}} = F_0$ (Definition 7.3.2.9).
- $(b)$
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a left Kan extension of $F_0$ along $\iota $ if there exists a natural transformation $\beta : F_0 \rightarrow F|_{ \operatorname{\mathcal{C}}^{0} }$ which exhibits $F$ as a left Kan extension of $F_0$ along $\iota $, in the sense of Variant 7.3.1.5.
These definitions are not quite equivalent. By virtue of Proposition 7.3.2.6, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfies condition $(a)$ if and only if it satisfies a stronger version of condition $(b)$, where $\beta $ is required to be an identity natural transformation. In particular, condition $(a)$ implies condition $(b)$. However, the converse is false: if $F$ is a left Kan extension of $F_0$ along $\iota $, then the restriction $F|_{ \operatorname{\mathcal{C}}^{0} }$ need not be equal to $F_0$. However, it is necessarily isomorphic to $F_0$, by virtue of Corollary 7.3.2.7.