Ah, ok, I see now that I should have used the function on objects induced by the profunctor \\(\cap_\mathcal{X}\\). So, the first diagram chase should say:

Let \\(x\in\mathcal{X}\\).

\[ \lambda_\mathcal{X}^{-1}(x)=1\otimes x.\]

\[ ( \cap\_\mathcal{X}\otimes1\_\mathcal{X})(1\otimes x)=(x\otimes x)\otimes x .\]

\[ \alpha_\mathcal{X,X^\mathrm{op},X}((x\otimes x)\otimes x)=x\otimes(x\otimes x). \]

\[ (1\_\mathcal{X}\otimes\cup\_\mathcal{X})(x\otimes(x\otimes x))=x\otimes1.\]

\[ \rho_\mathcal{X}(x\otimes\mathrm{id}_{x})=x.\]
, and the second one should be similarly modified.

I guess I was confused by the fact that I expected \\(\cap_\mathcal{X}(x)\\) to give the same result indepedent of \\(x\in\mathcal{X}\\), because I had in mind insertion of the identity operator in quantum mechanics.