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.