After reading Matthew's proof in [#8](https://forum.azimuthproject.org/discussion/comment/18540/#Comment_18540), I think that a \$$\mathbf{Cost}^{\text{op}}\$$-category is equivalent to a \$$(\\{ \tt{true}, \tt{false}\\}, \implies, \tt{false}, \tt{or}) \$$-category, that Daniel described in [#10 of Lecture 29](https://forum.azimuthproject.org/discussion/comment/18390/#Comment_18390).

Both Daniel and Matthew point out that either, \$$\mathcal X(x,y) = \tt{true}\$$ for all pairs \$$x,y)\$$ or \$$\mathcal X(x,y) = \tt{false}\$$ for all pairs \$$x,y)\$$. Thinking about the set of objects as a discrete points and drawing an arrow between objects where \$$\mathcal X(x,y) = \tt{true}\$$, we could visualize every \$$\mathbf{Cost}^{\text{op}}\$$-category as either a discrete set of points or as a fully connected graph.

Somehow this has the feeling of adjoints, since the discrete set of points under-estimates the connections between objects and the fully connected graph over-estimates the connections between objects.