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To a big -tilting object in a complete, cocomplete abelian category with an injective cogenerator we assign a big -cotilting object in a complete, cocomplete abelian category with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of and . Under various assumptions on , which cover a wide range of examples (for instance, if is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.