We have the following result:if $K:M\rightarrow C;\ T:M\rightarrow A$, and if the copower $[Tm',c]\cdot Tm$ exists, then $T$ has a Left Kan Extension given by $Lc=\int ^m[Km,c]\cdot Tm$ whenever the coend exists.
The proof is a straightforward application of the Yoneda Lemma, and the Fubini Theorem for coends. (MacLane, $X.4.2$). It amounts to a string of natural isomorphisms whose upshot is $[L,S] \cong [T,A^KS];\ S:C\rightarrow A$.
I have not been able to arrive at MacLane's formula for the unit $\eta $ of the adjunction by tracking the isomorphisms, but when I took another route and proved that
$\int ^m[Km,c]\cdot Tm=$colim $\left ( (K\downarrow c)\overset{P}{\rightarrow}M\overset{T}{\rightarrow}A \right )$, then I was able to show that the unit is given by
But I want to do it by chasing $1:L\rightarrow L$ through the isomorphisms above.