We introduce the notion of a Nakayama functor relative to an adjunction, generalizing the classical Nakayama functor for a finite-dimensional algebra. We show that it can be characterized in terms of an ambidextrous adjunction of monads and comonads. We also study this concept from the viewpoint of Gorenstein homological algebra. In particular we obtain a generalization of the equality of the left and right injective dimension for a finite-dimensional Iwanaga-Gorenstein algebra, and for a module category we show that this property can also be characterized by the existence of a tilting module.