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.