The bicategory is the locally full sub-2-category of determined by the maps.
In the bicategory Prof of categories and profunctors (perhaps enriched), if is a Cauchy complete category, then a profunctor is a map if and only if it is represented by a functor . If is not Cauchy complete, then maps correspond to functors from to the Cauchy completion of .
If every map in is comonadic? and has a terminal object, then is equivalent to a -category. If in addition is a cartesian bicategory and every comonad in has an Eilenberg--Moore object, then is biequivalent to , having finite limits. The converse is true if pullback squares in satisfy the Beck–Chevalley condition in , i.e. if their mates are invertible (see [LWW10]).
Similarly, is a topos if and only if is a unitary tabular power allegory.
A 2-category equipped with proarrows is, by definition, a bijective-on-objects pseudofunctor such that the image of every arrow in is a map in . Equivalently, therefore, it is a bijective-on-objects pseudofunctor .
Hence the inclusion is the “universal” proarrow equipment that can be constructed with a given bicategory as its bicategory of proarrows. More precisely, there is a forgetful functor from to which remembers only the bicategory of proarrows, and the assignment of to is its right adjoint.
Mike Shulman: This is obviously morally true, but I can’t be bothered right now to check which 1-, 2-, or 3-categories of equipments and bicategories one has to use to make it precisely correct.
A lot of work in bicategories that makes use of maps could easily be reformulated in a proarrow equipment, and conversely. Thus, it is to some extent a question of aesthetics which is preferred. One advantage of proarrow equipments is they can distinguish between a category and its Cauchy completion (as objects of Prof), while maps in bicategories are perhaps simpler in some ways.