  Relational structures are emerging as ubiquitous mathematical machinery in
the semantics of open systems of various kinds. Cartesian bicategories are a
well-known categorical algebra of relations that has proved especially useful
in recent applications. The passage between a category and its bicategory of
relations is an important question that has been widely studied for decades. We
study an alternative construction that yields a cartesian bicategory of
relations. Its behaviour is closely related to the axiom of choice, which
itself can be expressed in the language of cartesian bicategories.
