Background:
The limit of a functor $F:D^{op}\to C$ is an object $\lim F$ representing the functor $$\ell F(x):=\operatorname{Psh}_D(\ast,C(x, F(\cdot))),$$ where $*$ denotes the terminal presheaf on $D$. (Notice that $C(x, F(\cdot)))$ is a presheaf on $D$).
We can define the limit of a functor weighted by a presheaf in much the same way (by replacing $\ast$ with a fixed presheaf on $D$ called the weight).
Why am I bringing this up? It is a very slick definition. Nowhere do we have anything like universal arrows popping up. Adjunctions are out of sight and out of mind. Indeed, this definition generalizes straightforwardly to S-enriched categories for S symmetric monoidal closed (and all of the other requirements you need for the S-enriched Yoneda lemma to work).
The usual definition of the Kan extension is as a functor completing a certain commutative triangle such that it is universal in a specific sense in a certain functor category (intentionally vague...). This definition is pretty annoying to work with and is avoided whenever possible by instead insisting that all Kan extensions be pointwise (for instance, Kelly does this his book on enriched categories).
Question:
Does there exist a similar slick definition of the Kan extension (not necessarily pointwise)? By slick here, we mean free of adjoint functors (and their less conspicuous cousins, universal arrows) and free of commutative diagrams (translating the content of a commutative diagram into prose does not count).
