> These days, smart people are advocating the term **short map** for a map \$$F\$$ with
>
> $d'(F(x),F(x')) \le d(x,x').$

This is in line with my intuition - these are exactly "pairwise [Lipschitz continuous](https://en.wikipedia.org/wiki/Lipschitz_continuity) functions with constant 1".

Sorry I screwed up the order previously.

> The concept of \$$\mathcal{V}\$$-functor, introduced above, cleverly slips an inequality into the game. One might have tried demanding
>
> $\mathcal{X}(x,x') = \mathcal{X'}(F(x),F(x'))$
>
> and this would change the answers to "What is a \$$\mathbf{Bool}\$$-functor?" and "What is a \$$\mathbf{Cost}\$$-functor?" in interesting and instructive ways.

To answer "What is a \$$\mathbf{Bool}\$$-functor?", it is an [elementary embedding](https://ncatlab.org/nlab/show/elementary+embedding) of one preorder into another.

To answer "What is a \$$\mathbf{Cost}\$$-functor?", it is a generalization of an [isometry](https://en.wikipedia.org/wiki/Isometry) like you mentioned.