  In this article the notion of virtual double category (also known as
fc-multicategory) is extended as follows. While cells in a virtual double
category classically have a horizontal multi-source and single horizontal
target, the notion of augmented virtual double category introduced here extends
the latter notion by including cells with empty horizontal target as well.
  Any augmented virtual double category comes with a built-in notion of
"locally small object" and we describe advantages of using augmented virtual
double categories as a setting for formal category rather than 2-categories,
which are classically equipped with a notion of "admissible object" by means of
a yoneda structure in the sense of Street and Walters.
  An object is locally small precisely if it admits a horizontal unit, and we
show that the notions of augmented virtual double category and virtual double
category coincide in the presence of all horizontal units. Without assuming the
existence of horizontal units we show that most of the basic theory for virtual
double categories, such as that of restriction and composition of horizontal
morphisms, extends to augmented virtual double categories. We introduce and
study in augmented virtual double categories the notion of "pointwise"
composition of horizontal morphisms, which formalises the classical composition
of profunctors given by the coend formula.
