I suspect \\(\sum_x \sum_y \Phi(x,y) = \sum_y \sum_x \Phi(x,y)\\) is always true, since it comes from the associativity law of profunctors. \\(\sum_x \sum_y \Phi(x,y) = \sum_y \sum_x \Phi(x,y)\\) is just associativity along the idenities \\(I\_{\mathcal{X}}(x,x), I\_{\mathcal{Y}}(y,y)\\),

\\[
\sum_x \sum_y (I\_{\mathcal{X}}(x,x)\times \Phi(x,y))\times I\_{\mathcal{Y}}(y,y) \\\\
= \sum\_{x,y} I\_{\mathcal{X}}(x,x) \times \Phi(x,y)\times I\_{\mathcal{Y}}(y,y) \\\\
= \sum_y \sum_x I\_{\mathcal{X}}(x,x) \times (\Phi(x,y) \times I\_{\mathcal{Y}}(y,y))
\\]