Yes.
\$$d\$$ is the distance metric of whatever \$$\mathbf{Cost}\$$-category we happen to be working in (let's call it \$$\mathcal{C}\$$). It must satisfy the equations,

\$0 \geq d(x,x), \text{ for all } x\in Ob(\mathcal{C}), \\\\ d(x,y) + d(y,z) \geq d(x,z), \text{ for all } x,y,z \in Ob(\mathcal{C}), \\\\ \text{where } Ob(\mathcal{C}) = [0,\infty]. \$