> **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}\\).

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\\).

If \\(\mathcal{C}\\) is already a preorder, this doesn't do anything to it.

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\\).

If \\(\mathcal{C}\\) is already a preorder, this doesn't do anything to it.