Equalisers and coequalisers in categories with a \dag-involution

["Jamie Vicary" <jamievicary@gmail.com>]

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.