Keith - the middle diagram in your trio of diagrams can't equal the other two, because its top and bottom look different from the other two. It shows a feasibility relation going from 'nothing' (namely \$$\textbf{1}\$$) to \$$X \times X^{\text{op}}\$$ (or maybe \$$X^{\text{op}} \times X\$$, depending on your conventions) while the other two show feasibility relations going from \$$X\$$ to \$$X\$$.

Your equations don't make sense to me. Take \$$\text{hom}^{\text{op}}(x,x) \times 1_X\$$, for example. \$$1_X\$$ could be the name for a feasibility relation from \$$X\$$ to \$$X\$$, or it could be the name for a monotone function from \$$X\$$ to \$$X\$$. I don't know which since you didn't say which category these equations are describing morphisms in.

I don't know what \$$\text{hom}^{\text{op}}(x,x)\$$ means, or what particular value of \$$x\$$ you're talking about. No matter what guess I make, I'm having trouble seeing what \$$\text{hom}^{\text{op}}(x,x) \times 1_X\$$ is supposed to mean.

Hmm, maybe you are using \$$\text{hom}^{\text{op}}(x,x)\$$ as a name for the hom-functor for \$$X^{\text{op}}\$$. It that's what you mean, call it \$$\text{hom}\$$, or better \$$\text{hom}_{X^{\text{op}}}\$$ to indicate which preorder it's the hom-functor of.

There's a hom-functor

$\text{hom}\_X : X^{\text{op}} \times X \to \mathbf{Bool},$

and similarly a hom-functor

$\text{hom}_{X^{\text{op}}} : (X^{\text{op}})^{\text{op}} \times X^{\text{op}} \to \mathbf{Bool},$

or

$\text{hom}_{X^{\text{op}}} : X \times X^{\text{op}} \to \mathbf{Bool}$

for short.