Local reflections between relations, spans and polynomials

Given a locally cartesian closed regular category ξ we may form the bicategories of relations, spans and ploynomials. We show that for each hom-category, relations are a reflective subcategory of spans, and spans are a coreflective subcategory of ploynomials (with cartesian 2-cells). We then use these local reflections and coreflections to derive the universal property of relations from that of spans, and construct a right adjoint to the inclusion of spans into polynomials in the 2-category of bicategories, lax functors and icons. Moreover, we show that this right adjoint becomes a pseudofunctor if we restrict ourselves to polynomials for which the middle map is a monomorphism, or alternatively if we restrict ourselves to polynomials for which this map is a regular epimorphism.