Structured versus Decorated Cospans

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