The map $\ev$ is well-defined by Theorem 3.1 (i). It is a homeomorphism onto its image by Corollary 2.5 and the fact that any model is dominated by an SNC model. As $X$ is compact, we only need to show that $\ev(X)$ is dense in $\varprojlim\Delta_{\mathcal{X}}$. Pick $s=(s_{\mathcal{X}})_{\mathcal{X}}\in\varprojlim\Delta_{\mathcal{X}}$ and fix an SNC model $\mathcal{X}$. If $\mathcal{Y}$ is an SNC model dominated by $\mathcal{X}$, then $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}=\id$ yields $\ev_{\mathcal{Y}}(\emb_{\mathcal{X}}(s_{\mathcal{X}}))=s_{\mathcal{Y}}$. Hence $s=\lim_{\mathcal{X}}\ev(\emb_{\mathcal{X}}(s_{\mathcal{X}}))\in\overline{\ev(X)}$. ∎