Literature on Kan extensions Is there any literature on Kan extensions of functors whose domain is not a small category? Are there any general cases known when they exist? 
For example, say $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{C}\to\mathcal{E}$ are functors with one of them full and faithful and both $\mathcal{D}$ and $\mathcal{E}$ bicomplete (=complete+cocomplete) - do any of the Kan extensions exist?
 A: Given any right extension $HG\overset\epsilon\Rightarrow F\colon\mathcal C\to\mathcal E$ of $\mathcal C\xrightarrow{F}\mathcal E$ along $\mathcal C\xrightarrow{G}\mathcal D$, for each object $Y\in\mathcal D$ the morphisms $HY\xrightarrow{Hg}HGX\xrightarrow{\epsilon_X}FX$ indexed by objects $Y\xrightarrow{g}FX$ of the comma category $(Y\downarrow G)$ are the components of a cone with vertex $HY\in\mathcal E$ over the diagram $(Y\downarrow G)\xrightarrow{\Pi_G}\mathcal C\xrightarrow{F}\mathcal E$ where $\Pi_G(Y\xrightarrow{g}FX)=X$.
Furthermore, for any natural transformation $H_1\overset\phi\Rightarrow H_2\colon\mathcal D\to\mathcal E$ so that $H_1G\overset{\epsilon_1}\Rightarrow F\colon\mathcal C\to\mathcal E$ factors as $H_1G\overset{\phi G}\Rightarrow H_2G\overset{\epsilon_2}F\colon\mathcal C\to\mathcal E$, the morphism $H_1Y\xrightarrow{\phi_Y}H_2Y$ is a morphism between the cones on $(Y\downarrow G)\xrightarrow{\Pi_G}\mathcal C\xrightarrow{F}\mathcal E$. One can then show that a sufficient condition for the right Kan extension (i.e. terminal right extension) to exist is that for each object $Y\in\mathcal D$, the diagram $(Y\downarrow G)\xrightarrow{\Pi_G}\mathcal C\xrightarrow{F}\mathcal E$ has a limit, in which case a right Kan extension is given by choosing vertices of limiting cones of the diagrams. One might say that in this situation the right Kan extension is pointwise because its values at any object end up computable as limits that do not depend on values of the extension at other points (the values of pointwise left Kan extensions are analogous colimits).
This condition that the (co)limits of all those diagrams exist is not a necessary condition for Kan extension to exist, roughly because it is not the case that every cone on $(Y\downarrow G)\xrightarrow{\Pi_G}\mathcal C\xrightarrow{F}\mathcal E$ necessarily arises from some right extension as according to the above construction. 
However, in the case when $\mathcal E$ is locally small we can characterize the pointwise Kan extension using representable functor. Specifically, any pointwise Kan extension is preserved by each representable functor $\mathcal E\xrightarrow{Hom_{\mathcal E}(-,Z)}\mathcal Set$, and conversely any Kan extension preserved by each representable functor is pointwise. Thus, the condition that the (co)limits of all those diagrams exist is necessary and sufficient for the existence of pointwise Kan extensions.
