Kan extensions as coends and ends

Fix $d\in \mathcal{D}$. By [L1], the coend ${\int }^c\mathcal{D}(Kc,d)\cdot F(c)$ is the weighted colimit of $F$ by the presheaf $c\mapsto \mathcal{D}(Kc,d)$. By [F1], for every $m\in \mathcal{E}$ morphisms from this object to $m$ are in natural bijection with natural transformations $\mathcal{D}(K-,d)\Rightarrow \mathcal{E}(F-,m)$. Unwinding the two functors, such a natural transformation is exactly a family of maps $F(c)\to m$ indexed by arrows $u:Kc\to d$, natural in morphisms of $(K\downarrow d)$; that is precisely a cocone under the comma-category diagram at $d$. Therefore the coend has the same objectwise universal property as the pointwise left Kan extension value at $d$, so [L2] identifies them.

Again fix $d$. By [L1], the end ${\int }_{c}F(c{)}^{\mathcal{D}(d,Kc)}$ is the weighted limit of $F$ by the copresheaf $c\mapsto \mathcal{D}(d,Kc)$. By [F1], for every $m\in \mathcal{E}$ morphisms from $m$ to this object are in natural bijection with natural transformations $\mathcal{D}(d,K-)\Rightarrow \mathcal{E}(m,F-)$. Unwinding these data gives exactly a cone over the diagram on $(d\downarrow K)$ with vertex $m$. So this end has the same objectwise universal property as the pointwise right Kan extension value at $d$, and [L2] identifies them.

Since the argument is objectwise in $d$, the displayed coend and end formulas compute the values of the pointwise Kan extensions at every object of $\mathcal{D}$.