Fonctorial Construction of Frobenius Categories

Let $\Ascr,\Bscr$ be exact categories with $\Ascr$ karoubian and $M$ be an exact functor. Under suitable adjonction hypotheses for $M$, we are able to show that the direct factors of the objects of $\Ascr$ of the form $MY$ with $Y \in \Bscr$ make up a Frobenius category which allow us to define an $M$-stable category for $\Ascr$ only by quotienting. In addition, we propose a construction of an $M$-stable category for $\Ascr,\Bscr$ triangulated categories and $M$ a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck which links the two notions of $M$-stable category.