# Kerodon

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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 $F|_{ \operatorname{\mathcal{C}}^{0} }$.