Brown representability does not come for free

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

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