Brown representability does not come for free

We exhibit a triangulated category Τ having both products and coproducts, and a triangulated subcategory δ ⊂ Τ which is both localizing and colocalizing7 for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category δ nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets.