Re: Combining monads and Pullback cats

Mike is right. The problem lies in the existence of a left adjoint. However if one pulls back over a Grothendieck co-(=op-)fibration, then if U has a left adjoint, then so does its pullback. In fact, a functor is a cofibration iff it is universal for this property, i.e., it has this property and this property is stable under arbitrary pullbacks."A cofibration is a universal change of base for right adjoints". I don't know if this is of much help, but it is amusing.