  We establish the universal properties of the bicategory of polynomials,
considering both cartesian and general morphisms between these polynomials. A
direct proof of these universal properties would be impractical due to the
complicated coherence conditions arising from polynomial composition; however,
in this paper we avoid most of these coherence conditions using the properties
of generic bicategories.
  In addition, we give a new proof of the universal properties of the
bicategory of spans, and also establish the universal properties of the
bicategory of spans with invertible 2-cells; showing how these properties may
be used to describe the universal properties of polynomials.
