Let \({}_ RK\) be a left unitary module over a ring R with unity and \({\mathcal C}(_ RK)\) be the category of topological left R-modules, isomorphic to closed submodules of powers of \({}_ RK\) endowed with the discrete topology. For \(A=End(_ RK)\) consider the category Mod-A of right unitary A-modules and its subcategory \({\mathcal D}(K_ A)\) cogenerated by \(K_ A\). The bimodules \({}_ RK_ A\) which generate (in the standard way) a duality between \({\mathcal C}(_ RK)\) and \({\mathcal D}(K_ A)\) such that the K-valued continuous characters in \({\mathcal C}(_ RK)\) can be extended where characterized by \textit{C. Menini} and \textit{A. Orsatti} [Ann. Mat. Pura Appl., IV Ser. 127, 187-230 (1981; Zbl 0476.16029)]. The aim of the paper under review is to describe the particular case when \({\mathcal D}(K_ A)=Mod\)-A, i.e. \(K_ A\) is a cogenerator of Mod-A. The author shows that this occurs precisely when \({\mathcal C}(_ RK)\) is an abelian category. Using results of \textit{A. Orsatti} and the reviewer [Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat. 8, 143-183 (1984; Zbl 0562.16023)] he obtains various constraints on the structure of \({}_ RK\). A detailed characterization is given in the more particular case when \(K_ A\) is an injective cogenerator of Mod-A. Finally, an example of a duality of the above type such that \(K_ A\) is a non-injective cogenerator of Mod-A is provided.