>**Puzzle 89.** Figure out exactly what a \$$\mathbf{Cost}\$$-category is, starting with the definition above. Again, what matters is not your final answer so much as your process of deducing it!

>Hint: given a \$$\mathbf{Cost}\$$-category \$$\mathcal{X}\$$, write \$$\mathcal{X}(x,y)\$$ as \$$d(x,y)\$$ and call it the **distance** between the objects \$$x,y \in \mathrm{Ob}(\mathcal{X})\$$. The axioms of an enriched category then say interesting things about "distance".

Both the book (on page 53) and [Simon Willerton, on his Catester videos on enriched categories](https://www.youtube.com/watch?v=be7rx29eMr4&list=PL398472E8DD6B73C8), spill the beans that such a \$$\mathbf{Cost}\$$- or \$$\mathbb{R}_+ \$$-category is a metric space.

Now I'm not sure if *proof by book* or *proof by Catster* will count, but you did want us to get creative with the proving process.

Anyways, joking aside... For those wanting to give a proof, I would recommend reading *'Seven Sketches'* and watching that video playlist.