On a theorem of Brown and Adams

Let ℸ be the homotopy category of all spectra, Tc ⊂ T the full sub-category of finite spectra. Brown and Adams proved that any homological functor H:Tcop → Ab is the restriction of a representable functor on T. Furthermore, any natural transformation is the restriction of a map in ℸ. y naturally wonder whether this generalizes to arbitrary triangulated categories, for instance D(R) where R is a ring. We show that the answer is in general no, although for R countable the generalization holds.