Pointwise Kan extensions as those preserved by representables

Let $K:\mathcal{C}\to \mathcal{D}$ and $F:\mathcal{C}\to \mathcal{E}$ be functors, with $\mathcal{D}$ and $\mathcal{E}$ locally small (Small, locally small, and large categories).

Suppose $(R,\epsilon )$ is a right Kan extension of $F$ along $K$ (Left and right Kan extensions). It is pointwise when, for every object $e$ of $\mathcal{E}$, the covariant representable functor

$\mathcal{E}(e,-):\mathcal{E}\to \mathbf{S}\mathbf{e}\mathbf{t}$

(The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category) sends $(R,\epsilon )$ to a right Kan extension of $\mathcal{E}(e,F-)$ along $K$.

Suppose $(L,\eta )$ is a left Kan extension of $F$ along $K$. It is pointwise when, after passage to opposite categories (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$), the corresponding right Kan extension in ${\mathcal{E}}^{\mathrm{o}\mathrm{p}}$ is preserved by every covariant representable of ${\mathcal{E}}^{\mathrm{o}\mathrm{p}}$, equivalently by every contravariant representable

$\mathcal{E}(-,e):{\mathcal{E}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}\mathrm{.}$

So the phrase "preserved by representables" is not one definition but two variance-specific ones: covariant representables for right Kan extensions, and contravariant representables after passing to opposites for left Kan extensions.