  Hilsum-Skandalis maps, from differential geometry, are studied in the context
of a cartesian category. It is shown that Hilsum-Skandalis maps can be
represented as stably Frobenius adjunctions. This leads to a new and more
general proof that Hilsum-Skandalis maps represent a universal way of inverting
essential equivalences between internal groupoids. To prove the representation
theorem, a new characterisation of the con- nected components adjunction of any
internal groupoid is given. The charaterisation is that the adjunction is
covered by a stable Frobenius adjunction that is a slice and whose right
adjoint is monadic. Geometric morphisms can be represented as stably Frobenius
adjunctions. As applications of the study we show how it is easy to recover
properties of geometric morphisms, seeing them as aspects of properties of
stably Frobenius adjunctions.
