I think I am not understanding something correctly about \\(\mathbf{Cost}\\)-categories. Isn't the composition rule for \\(\mathbf{Cost}\\)-categories like this: If we have objects \\(N,W,S\\), then the composition rule is \\(\mathcal{C}(S,W) + \mathcal{C}(W,N) \ge \mathcal{C}(S,N)\\) so \\(4 \ge \mathcal{C}(S,N)\\). There's another path between \\(S\\) and \\(N\\) via \\(E\\), but it gives \\(7 \ge \mathcal{C}(S,N)\\). Does this mean that \\(\mathcal{C}(S,N) = 4\\) since it's the meet of compositions that give \\(\mathcal{C}(S,N)\\)?

Also, it seems to me that \\(\mathcal{V}\\)-profunctors are a way of gluing together two \\(\mathcal{V}\\)-categories such that the result is a \\(\mathcal{V}\\)-category. Like for two \\(\mathbf{Bool}\\)-categories (say \\(\mathcal{X}\\) and \\(\mathcal{Y}\\)) and a feasibility relation \\(\Psi\\), we have a \\(\mathbf{Bool}\\)-category \\(\mathcal{Z}\\) where \\(\mathrm{ob}(\mathcal{Z}) = \mathrm{ob}(\mathcal{X}) \sqcup \mathrm{ob}(\mathcal{Y}) \\) and \\(\mathcal{Z}(a,b)\\) is true when either \\(\mathcal{X}(a,b)\\) or \\(\mathcal{Y}(a,b)\\) or \\(\Psi(a,b)\\) is true (basically when there's a path \\(a\\) to \\(b\\)). Is that intuition right?

Also, it seems to me that \\(\mathcal{V}\\)-profunctors are a way of gluing together two \\(\mathcal{V}\\)-categories such that the result is a \\(\mathcal{V}\\)-category. Like for two \\(\mathbf{Bool}\\)-categories (say \\(\mathcal{X}\\) and \\(\mathcal{Y}\\)) and a feasibility relation \\(\Psi\\), we have a \\(\mathbf{Bool}\\)-category \\(\mathcal{Z}\\) where \\(\mathrm{ob}(\mathcal{Z}) = \mathrm{ob}(\mathcal{X}) \sqcup \mathrm{ob}(\mathcal{Y}) \\) and \\(\mathcal{Z}(a,b)\\) is true when either \\(\mathcal{X}(a,b)\\) or \\(\mathcal{Y}(a,b)\\) or \\(\Psi(a,b)\\) is true (basically when there's a path \\(a\\) to \\(b\\)). Is that intuition right?