>EDIT to add: in which case I think this line is wrong:

>> \[ (\mathcal{X}^{\text{op}} \times \mathcal{X})((a,b),(c,d)) = \mathcal{X}(b,a) \otimes \mathcal{X}(c,d) \]

> It should read
>\[ (\mathcal{X}^{\text{op}} \times \mathcal{X})((a,b),(c,d)) = \mathcal{X}^\text{op}(a, c) \otimes \mathcal{X}(b, d) = \mathcal{X}(c, a) \otimes \mathcal{X}(b, d) \]
>... and then the proof drops out.

Thank you so much for taking the time to review my attempted proof!

I hope it's okay, I've gone back and corrected my proof based on your comments.