Although I cannot be sure, this looks an awful lot like an adjoint triple. I think Charles and I had a section of TTT on this. It was certainly not new with us and may have even been in Harry Appelgate's thesis back around 40 years ago. Michael On Thu, 20 May 2004, Gaunce Lewis wrote:
I have encountered a situation in which I have two categories C, D which are related by a pair of adjoint functors L from C to D and R from D to C. Also, there is a cotriple S on C and a cotriple T on D. Finally, there is a natural isomorphism f from RT to SR. It seems that if a couple of diagrams relating f to the structure maps of the cotriples commute, then there is an induced adjoint pair relating the two coalgebra categories. Is this, or something similar to it, in the literature in some easily referenced place?
Thanks, Gaunce