[Anindya wrote](https://forum.azimuthproject.org/discussion/comment/18393/#Comment_18393):
> Surely \$$\mathcal{X}(x, y)\$$ should be \$$\tt{true}\$$ if \$$x \mid y\$$ and \$$\tt{false}\$$ otherwise.

Aha, I misread Property 1! It says that \$$I \leq \mathcal{X}(x,x)\$$, but I swapped one \$$x\$$ out for a \$$y\$$, which would mean that everything has to be less than or equal to \$$\mathrm{true}\$$ if we take that as the identity. But as you've made me aware, that's not the case -- only \$$\mathcal{X}(x, x)\$$ is so constrained, and this corresponds to how categories require the presence of all identity arrows.

Let me see if I can recover this once more!