**Puzzle 103.** Figure out the best way to take any category \$$\mathcal{C}\$$ and "squash it down", getting a preorder whose elements are the objects of \$$\mathcal{C}\$$.

Matthew wrote:

> I would define \$$\leq_{\mathcal{C}}\$$ with:

> $X \leq_{\mathcal{C}} Y \equiv \mathrm{Hom}(X,Y) \neq \varnothing$

> where \$$\mathrm{Hom}(X,Y)\$$ is the [set of morphisms](https://en.wikipedia.org/wiki/Morphism#Hom-set) between \$$X\$$ and \$$Y\$$.

1. Given a category, create a preorder by decreeing \$$x \le y\$$ iff there's a morphism from \$$x\$$ to \$$y\$$.
2. Given a category, identify all morphisms from \$$x\$$ to \$$y\$$ - that is, decree them all to be equal - so that there's at most one morphism from \$$x\$$ to \$$y\$$.
These are equivalent because we can think of a preorder as a category with at most one morphism from any object \$$x\$$ to any object \$$y\$$: we write \$$x \le y\$$ iff such a morphism exists.