I was recently reading about Kan extensions and the codensity monad. Apparently you can think of the codensity monad as a bit like an endomorphism monoid internal to a functor category.
The right Kan extension is a bit like an exponential in a functor category.
$$[F, G](a) = \forall b, (a \rightarrow F(b)) \rightarrow G(b) $$
And you can "eval" and "curry" things as well.
$$\text{eval} : [F, G] \circ F \Rightarrow G$$
Left kan extension would be a "coexponential" like object. You can think of it as a little like a tuple of a value and a continuation.
$$(F - G)(a) = \Sigma_b\, F(b) \times (G(b) \rightarrow a) $$
It strikes me "exponential" isn't quite the right terminology. Composition certainly isn't a Cartesian product.
So left and right Kan extensions would be coresiduals and residuals?
I'm interested in exploring further so I want to know the precise and accurate terminology and notation instead of the hodgepodge of "its a bit like an exponential" I have right now.