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!