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.