Dear all, Consider the following straightforward coequaliser (e,E) formed by f,g:A-->B and e:B-->E, with e.f=e.g. I am working in a category with biproducts, and with a contravariant involutive endofunctor (--)^\dag on the category which is compatible with the biproducts; i.e. (projection)^\dag = injection for all projections and injections making up a part of a biproduct. In such a category, it is natural to consider the coequaliser object E to be the subspace of B on which the morphisms f and g agree. It is therefore natural to require e.(e^\dag) = id_E; this sort of condition is similar to the sorts of conditions that form part of the definition of the biproduct. I'm asking whether there exists a natural framework generalising the theory of biproducts, which is analagous to the way that (co)limits generalise (co)products, within which I can safely assume that e.(e^\dag) = id_E. Biproducts seem quite different from products and coproducts, though, so I don't know how it would work. Jamie Vicary.