The proof that it is a great title: I only heard about this by skimming a tiny workshop announcement I got in my email this morning and seeing the title, and figured I just had to read it.
---
The thing that surprises me just a little is the blatant asymmetry of definition 2: a relative adjunction between the functor J : J → C and the category D is given by two functors L : J → D and R : D → C such that there is a natural isomorphism C(JX, RY) = D(LX, Y).
What happens if we instead have a "doubly-relative" adjunction that has
J : J → C
L : J → D
R : K → C
M : K → D
such that there is a natural isomorphism
C(JX, RY) = D(LX, MY)
?
Surely we ought to get some relative comonads out of this, too, at the very least.
---
Oh, oops, Arrows are actually strong relative monads on the yoneda embedding. That explains all the fiddly axioms about products and projections and stuff I guess!
