  In this note, we provide an axiomatic framework that characterizes the stable
$\infty$-categories that are module categories over a motivic spectrum. This is
done by invoking Lurie's $\infty$-categorical version of the Barr--Beck
theorem. As an application, this gives an alternative approach to R\"ondigs and
\O stv\ae r's theorem relating Voevodsky's motives with modules over motivic
cohomology, and to Garkusha's extension of R\"ondigs and \O stv\ae r's result
to general correspondence categories, including the category of Milnor-Witt
correspondences in the sense of Calm\`es and Fasel. We also extend these
comparison results to regular Noetherian schemes over a field (after inverting
the residue characteristic), following the methods of Cisinski and D\'eglise.
