  Interest in weak cubical n-categories arises in various contexts, in
particular in topological field theories. In this paper, we describe a concept
of double bicategory, namely a strict model of the theory of bicategories in
Bicat. We show that in a special case one can reduce this to what we call a
Verity double bicategory, after Domenic Verity. This is a weakened version of a
double category, in the sense that composition in both horizontal and vertical
directions satisfy associativity and unit laws only up to (coherent)
isomorphisms. We prove that there are examples in the form of double
bicategories of "double cospans" (or "double spans") in any category with
pushouts (pullbacks, respectively), and give a construction from this which
involves taking isomorphism classes of objects, and gives a Verity double
bicategory of double cospans. We describe cobordisms with corners as an example
of this construction.
