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.