Suppose given a well-generated triangulated category with a compatible symmetric monoidal structure, $\mathcal{T}$ (in the sense of Neeman). Is it clear that the image of a localization functor will also be well-generated? If not in true in general, is it easy to prove with respect to the $X$-acyclics (i.e. objects which tensor with $X$ to zero) for some element $X$?

The main thing I'm worried about I suppose (though the second axiom for well-generated-ness is not obvious to me either) is the requirement for the generators to be $\alpha$-small for some cardinal $\alpha$. For instance, the $MU$-localization of the stable homotopy category has no small objects. But perhaps they are $\alpha$-small for some $\alpha$? My understanding of such issues is still somewhat superficial.