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.