@FredrickEisele

The metric function and the monoid function are different things, they have distinct domains.

The metric function takes two points (objects) in the almost metric space, and produces a element of \\( \[0,\inf\]\\).

The monoid takes two distances (elements of \\( \[0,\inf\]\\)) and produces a third.

This and the order on Cost determines what the "enriched" transivity means.