  Auslander and Kleiner proved in 1994 an abstract version of Green
correspondence for pairs of adjoint functors between three categories. They
produce additive quotients of certain subcategories giving the classical Green
correspondence in the special setting of modular representation theory.
Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the
classical setting of modular representation theory is actually an equivalence
between triangulated categories with respect to a non standard triangulated
structure. In the present note we first define and study a version of relative
projectivity, respectively relative injectivity with respect to pairs of
adjoint functors. We then modify Auslander Kleiner's construction such that the
correspondence holds in the setting of triangulated categories.
