Could we make the following definition: a dagger-category has 'finite bilimits' if any finite diagram D in the category has an 'isometric cone', a cone for which all the associated morphisms to the objects of D are isometries, along with some sort of condition that the isometries are orthogonal in the correct way. It is interesting to consider this in the case of products and equalisers: for products AxB, so it seems, the isometries to A and B will generally be _projectors_, but for equalisers E-e->A=f,g=>B, the isometry e will generally be an _injector_! So we cannot ask for the cone morphisms to be isometric projectors, or isometric injectors. But perhaps this is OK, and we can just require them to be isometries. This definition of bilimit has the 'local flavour' of the definition of biproducts, but cooking up a generally-applicable orthogonality condition on the isometries seems tricky.
Fred Linton has pointed out to me that my terminology here is not standard. By "isometric injector", I mean a morphism which is unitary on its range, i.e., one-to-one and norm-preserving in the case of Hilbert spaces; I believe this is usually simply referred to as an isometry. By "isometric projector", I mean a morphism which is unitary on the complement of its kernel; some people prefer to call this a "partial isometry". I was then using the terms "isometric" and "isometry" to mean "isometric projector or isometric injector". Anyway, the simple prescription I give for a bilimit cannot work, as it is easy to find diagrams in the category of finite-dimensional Hilbert spaces, our canonical example of a strongly compact-closed category with biproducts, for which the colimit and limit are not isomorphic. A diagram f:A-->B for non-iso A and B is the simplest example. However, if we restrict to diagrams F:D-->FdHilb such that D admits a dagger-operation compatible with the dagger on FdHilb, then I believe the conjecture becomes plausible. Regards, Jamie Vicary.