  One goal of applied category theory is to understand open systems. We compare
two ways of describing open systems as cospans equipped with extra data. First,
given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is
a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If
$\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ preserves them, it
is known that there is a symmetric monoidal double category whose objects are
those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans.
Second, given a pseudofunctor $F \colon \mathsf{A} \to \mathbf{Cat}$, a
"decorated cospan" is a diagram in $\mathsf{A}$ of the form $a \rightarrow m
\leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong,
we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+)
\to (\mathsf{Cat},\times)$ is symmetric lax monoidal, there is a symmetric
monoidal double category whose objects are those of $\mathsf{A}$ and whose
horizontal 1-cells are decorated cospans. We prove that under certain
conditions, these two constructions become isomorphic when we take $\mathsf{X}
= \int F$ to be the Grothendieck category of $F$. We illustrate these ideas
with applications to electrical circuits, Petri nets, dynamical systems and
epidemiological modeling.
