Let $\varphi\in\PSH(X,\theta)$. Then, as $\mathcal{X}$ runs through the directed set of SNC models on which $\theta$ is determined, $(\varphi\circ p_{\mathcal{X}})_{\mathcal{X}}$ forms a decreasing net of continuous functions on $X$, converging pointwise to $\varphi$.