Contributions to Brown Representability Problem

The formulation of one or another incarnation of Brown representability theorem may be technical, but the idea behind all of them is simple and elegant: We want to be able to construct right or left adjoints for triangulated functors preserving coproducts, respectively products; because the categories we work with do not satisfy the hypothesis of Freyd’s Adjoint Functor Theorem we need a replacement for it in the new setting. This is Brown representability.