  Let $\mathcal C$ be a category with finite colimits, writing its coproduct
$+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We
describe a method of producing a symmetric monoidal category from a lax braided
monoidal functor $F: (\mathcal C,+) \to (\mathcal D, \otimes)$, and of
producing a strong monoidal functor between such categories from a monoidal
natural transformation between such functors. The objects of these categories,
our so-called `decorated cospan categories', are simply the objects of
$\mathcal C$, while the morphisms are pairs comprising a cospan $X \rightarrow
N \leftarrow Y$ in $\mathcal C$ together with an element $1 \to FN$ in
$\mathcal D$. Moreover, decorated cospan categories are multigraph
categories---each object is equipped with a special commutative Frobenius
monoid---and their functors preserve this structure.
