A generalization of the Nakayama functor

In this paper we introduce a generalization of the Nakayama functor for finite-dimensional algebras. This is obtained by abstracting its interaction with the forgetful functor to vector spaces. In particular, we characterize the Nakayama functor in terms of an ambidextrous adjunction of monads and comonads. In the second part we develop a theory of Gorenstein homological algebra for such Nakayama functor. We obtain analogues of several classical results for Iwanaga-Gorenstein algebras. One of our main examples is the module category $\Lambda\text{-}\operatorname{Mod}$ of a $k$-algebra $\Lambda$, where $k$ is a commutative ring and $\Lambda$ is finitely generated projective as a $k$-module.