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

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

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