Exercise 7.3.0.2 (Uniqueness of Kan Extensions). Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors between categories, and suppose that $F$ is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Show that the restriction map
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Natural transformations from $F$ to $G$} \} \ar [d] \\ \{ \textnormal{Natural transformations from $F|_{\operatorname{\mathcal{C}}^{0}}$ to $G|_{\operatorname{\mathcal{C}}^{0} }$} \} } \]
is a bijection. In particular, the functor $F$ can be recovered (up to canonical isomorphism) from the restriction $F|_{ \operatorname{\mathcal{C}}^{0} }$.