We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the 2-cells are the maps of bimodules satisfying two simple compatibilities. The operation of getting the composed coring from a given entwining, is promoted here to a canonical morphism of bicategories from a bicategory of entwinings to the Streets bicategory of corings.
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