Day convolution for monoidal bicategories

Ends and coends can be described as objects which are universal amongst extranatural transformations. We describe a cate- gorification of this idea, extrapseudonatural transformations, in such a way that bicodescent objects are the objects which are universal amongst such transfor- mations. We recast familiar results about coends in this new setting, providing analogous results for bicodescent objects. In particular we prove a Fubini theorem for bicodescent objects. The free cocompletion of a category C is given by its category of presheaves [C^op ,Set]. If C is also monoidal then its category of presheaves can be pro- vided with a monoidal structure via the convolution product of Day. This monoidal structure describes [C^op ,Set] as the free monoidal cocompletion of C. Day’s more general statement, in the V-enriched setting, is that if C is a promonoidal V-category then [C^op ,V] possesses a monoidal structure via the convolution product. We define promonoidal bicategories and go on to show that if A is a promonoidal bicategory then the bicategory of pseudofunctors Bicat(A^op ,Cat) is a monoidal bicategory.