An adjunction between categories, a weakened notion of an equivalence, is omnipresent in mathematics. In particular, it provides a useful framework for studying various phenomena in algebraic topology. In this talk, I will introduce a method for constructing an adjunction of module categories. Given an algebra R and a left ideal J, the eigenring, introduced by Ore in 1932, is defined as a canonical subquotient algebra of R with respect to J. Recently, I gave a construction of an adjunction between the category of R-modules and the category of modules over the eigenring. By anology between monads and algebras, I also introduced the notion of eigenmonad and established a similar adjunction. This adjunction has interesting applications to polynomial functors, outer functors (on free groups), and the Habiro-Massuyeau category in quantum topology. In part I, I will present the general construction with some classical examples and an overview of the applications. In part II, I will give a detailed explanation of the applications to polynomial functors and the Habiro-Massuyeau category.