  Structured and decorated cospans are broadly applicable frameworks for
building bicategories or double categories of open systems. We streamline and
generalize these frameworks using central concepts of double category theory.
We show that, under mild hypotheses, double categories of structured cospans
are cocartesian (have finite double-categorical coproducts) and are equipments.
The proofs are simple as they utilize appropriate double-categorical universal
properties. Maps between double categories of structured cospans are studied
from the same perspective. We then give a new construction of the double
category of decorated cospans using the recently introduced double Grothendieck
construction. Besides its conceptual value, this reconstruction leads to a
natural generalization of decorated cospans, which we illustrate through an
example motivated by statistical theories and other theories of processes.
