  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.
