Show that a skeletal dagger Cost-category is an extended metric space.
We already know that a Cost-category is a Lawvere metric space. We only need to prove that the distance function of such metric space is symmetric and that implies .
The first condition is proved by appealing to the dagger condition. Since is a Cost-functor, . Dually, . Thus, since the ordering on the extended real is a total order, we can deduce that . This is true for all objects and in .
The second condition is proved by appealing to skeletality. Suppose . Symmetry tells us that . Then, and . Then, by the skeletality condition.