Hmm. Your proof as written doesn't quite work because you have \\(\chi (a,b)\\) where it should be \\(\chi(a,b)\\), and there's only guaranteed to be a maximal hom object if \\(Obj \times Obj\\) is finite.

But I think both issues can be repaired:

Let \\(a, b\\) be objects of a \\(\mathbf {Cost^{op}}\\)-category. By the co-triangle inequality

\\[\mathcal{X}(a,b)+\mathcal{X}(b,a)≤\mathcal{X}(a,a) \\]

Adding a non-negative number can't decrease a real value, so

\\[\mathcal{X}(a,b)≤\mathcal{X}(a,a) ,\\]

however by the co- triangle inequality

\\[ \mathcal{X}(a,b)+\mathcal{X}(a,a)≤\mathcal{X}(a,b) \\]

by\\(\mathbf {Cost^{op}}\\) being a monoidal preorder

\\[ \mathcal{X}(a,b)+\mathcal{X}(a,b)≤\mathcal{X}(a,b) \\]

\\[ 2\mathcal{X}(a,b)≤\mathcal{X}(a,b) \\]

So \\(\mathcal{X}(a,b)= 0 \\) or \\(\infty \\).