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

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


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

for short.