  A pair of biadjoint functors between two categories produces a collection of
elements in the centers of these categories, one for each isotopy class of
nested circles in the plane. If the centers are equipped with a trace map into
the ground field, then one assigns an element of that field to a diagram of
nested circles. We focus on the self-adjoint functor case of this construction
and study the reverse problem of recovering such a functor and a category given
values associated to diagrams of nested circles.
