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 \$$.