In a Lawvere metric space, \$$I=0\$$. So, an identity element is a map \$$\mathrm{id}_x:0\to\mathcal{X}(x,x)\$$. In terms of distances, this means that \$$0\geq d(x,x)\$$ (since \$$0\$$ is the smallest element of \$$[0,\infty]\$$, this means \$$d(x,x)=0\$$ (this was already observed in the text between the definition of a Lawvere metric space as a **Cost**-category and the example of \$$\mathbb{R}\$$ as a Lawvere metric space, in chapter 2), which is the statement of definiteness for a Lawvere metric).