By definition of $c_{\mathcal{X}}$ we have $c_{\mathcal{X}}(x)\in E_{i}$ for a given $i\in I$ iff $\langle\ev_{\mathcal{X}}(x),E_{i}\rangle>0$, and it follows that $\ev_{\mathcal{X}}(x)$ lies in the relative interior of the simplex $\sigma_{J}$ for the maximal $J\subset I$ such that $c_{\mathcal{X}}(x)\in E_{J}$. Property (b) in Theorem 3.1 then shows that $c_{\mathcal{X}}(p_{\mathcal{X}}(x))$ is the generic point of $E_{J}$, which proves (i).

Let us prove (ii). For each $x\in X$ we have

using the identity $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}=\id$.

∎