Pointwise Kan extensions by the comma-category formula

Let $K:\mathcal{C}\to \mathcal{D}$ and $F:\mathcal{C}\to \mathcal{E}$ be functors.

Suppose $(L,\eta )$ is a left Kan extension of $F$ along $K$ (Left and right Kan extensions). It is pointwise when, for every object $d$ of $\mathcal{D}$, the value $L(d)$ is computed by the comma-category colimit of Comma-category limit and colimit formulae compute Kan extensions: the family of morphisms

$L(u)\circ {\eta }_c:F(c)⟶L(d)$

indexed by the objects $(c,u:Kc\to d)$ of $(K\downarrow d)$ (Comma category, slice category, and coslice category) is a colimit cocone of the diagram $(K\downarrow d)\to \mathcal{C}\to \mathcal{E}$. In particular, at $d=Kc$ and $u=1_{Kc}$ this leg is ${\eta }_c$.

Suppose instead that $(R,\epsilon )$ is a right Kan extension of $F$ along $K$. It is pointwise when, for every object $d$ of $\mathcal{D}$, the value $R(d)$ is computed by the comma-category limit formula: the family

${\epsilon }_c\circ R(u):R(d)⟶F(c)$

indexed by the objects $(c,u:d\to Kc)$ of $(d\downarrow K)$ is a limit cone of the diagram $(d\downarrow K)\to \mathcal{C}\to \mathcal{E}$. In particular, at $d=Kc$ and $u=1_{Kc}$ this leg is ${\epsilon }_c$.