If $\varphi\in\PSH(X,\theta)$ and $\mathcal{X}$ is an SNC model on which $\theta$ is determined, then:

• (i)

  $\varphi\circ\emb_{\mathcal{X}}$ is continuous on $\Delta_{\mathcal{X}}$ and convex on each face;

• (ii)

  $\varphi\circ p_{\mathcal{X}}$ is continuous on $X$ and $\varphi\leq\varphi\circ p_{\mathcal{X}}$.