Just thinking out loud here...

If we define a function called \\(\mathrm{when}\\),

\\[
\mathrm{when}(b,x) :\mathbf{Bool}\otimes \mathcal{X} \to \mathcal{X} \\\\
:= \text{if } b \text{ then } x \text{ else } \varnothing,
\\]

then \\(\mathrm{when}\\) is adjoint (in fact equivalent to) to taking the Cartesian product with a Boolean variable,

\\[
(b \times -) \vdash \mathrm{when}(b,-).
\\]

So then a \\(\mathbf{Cost}\\)-morphism is something like \\(\mathrm{when}(x\multimap y,r)\\), ie are \\(\mathbf{Bool}\\)-morphisms that are labeled by a real number.