Cost categories are Lawvere metric spaces and can be represented by weighted graphs
This is a depiction of a Cost-profunctor
The distance from a point in the left to a point in the right will run through the left, across a bridge, then through through the right.
\(\phi(B,x)=\)\(11\),\(\phi(A,z)=\)\(20\),\(\phi(C,y)=\)\(17\)