[red-herring comment](https://forum.azimuthproject.org/discussion/comment/18489/#Comment_18489)
Funny.
The poset vs. preordered set vs. partially-order set, discussion that happened earlier seems to be a case of the mathematical red herring principle almost happening.

When we start with a set, let's call it \$$\mathbb{X} \$$ and some relation \$$\oplus \$$ we do this...
$$\mathcal{X} := ( \mathbb{X}, \oplus )$$
...and then apply some rules governing the behavior of \$$\oplus \$$ over \$$\mathbb{X} \$$.
What do we call what was just done and what was formed namely \$$\mathcal{X} \$$?

One more question.
Is this a case where all \$$\mathcal{V}\text{-categories} \$$ are \$$\text{categories} \$$ but not all \$$\text{categories} \$$ are \$$\mathcal{V}\text{-categories} \$$?
Surely there will be some relationship between them [maybe I should just be patient and all will be revealed].