Isomorphism of Left Kan extensions

Question

I just got introduced to Kan Extensions in my Introduction to Category Theory course and I am trying to solve this exercise$:$ Let $K:\mathcal{A} \rightarrow \mathcal{C}$ be a functor. Let $A \in \mathcal{A}$ and $F=\mathcal{A}(A,-):\mathcal{A} \rightarrow Set$. Show that $Lan_K \mathcal{A}(A,-)$ is isomorphic to $\mathcal{C}(KA,-)$ with unit $\mathcal{A}(A,-) \Rightarrow \mathcal{C}(KA,K-)$ corresponding via Yoned to $id_{KA}$. I was trying to build a map: $Nat(\mathcal{A}(-,A),LK) \rightarrow \mathcal{C}(KA,K_)$ sending a morphism $\alpha$ to $\alpha _B(f)$, like in the proof of Yoneda lemma , and I think it should work but I don't know how to proceed. Any help would be highly appreciated.

Answer 1

For any functor $G\colon \mathcal B \to \mathrm{Set}$ you have natural bijections $$ \mathrm{Nat}(\mathcal A(A,-), G\circ K) \cong G(KA) \cong \mathrm{Nat}(\mathcal B(KA,-), G) $$ which shows that $\mathcal B(KA,-)$ satisfies the universal property of the left Kan extension.

Can you reconstruct the unit from this?