Exercise X.3.2 Mac Lane CWM (Kan extension of representable functors) I found an exercise on Mac Lane CWM , pg.240 ex.2 :
If A= $Set$, and M, C have small hom-sets, show that the left Kan extension of $M$(m,-) is $C$(Km,-) with unit $\eta$:$Id_M$ $\rightarrow$$C$(Km,K-) given by $\eta$m=$1_{Km}$   (with K:M$\rightarrow$C)
I tried to compute it directly by the given formula in the book: 
L(c)=$Colim$((K$\downarrow$c)
$\rightarrow$ M$\rightarrow$A) without success.Is it possible to show it in this way, or, do you know other methods?
Thank you!

Here is a sketch of the proof:
(I). Define $\eta _{[m,m']}:[m,m']\to [Km,Km']$ in the obvious way $f\mapsto Kf$ and show it is natural. 
(II). Suppose $\alpha:[m,-]\overset{\cdot }{\rightarrow} A^KS=SK$ is a given natural transformation. Then, Yoneda applies to show that there is a unique $a\in SKm$ such that $\alpha_{m'}(f:m\to m')=SKf(a).$ 
(III). Define $\sigma: [Km,-]\overset{\cdot }{\rightarrow}S$ to be $\sigma_c(g:Km\to c)=Sg(a)$ where $a\in SKm$ is as in (II). 
(IV). Finally, show that $A^K\sigma\circ \eta _{[m,-]}=\alpha$ by considering components $m'$ and using (II) and (III). Uniqueness of $\sigma $ follows from Yoneda. 
