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?