Upon considering the new family $\varphi_{I}=\max_{\alpha\in I}\varphi_{\alpha}$ with $I$ ranging over all finite subsets of $A$, we may assume that $A$ is a directed set and $(\varphi_{\alpha})$ is an increasing net. For each SNC model $\mathcal{X}$ we have $\varphi_{\alpha}\leq\varphi_{\alpha}\circ p_{\mathcal{X}}$ for all $\alpha$, hence $\varphi\leq\varphi\circ p_{\mathcal{X}}$. By Corollary 7.7 $\varphi_{\alpha}\circ i\mathcal{X}$ converges uniformly to $\varphi\circ i_{\mathcal{X}}$, which is therefore continuous. Using Lemma 7.10 we conclude that $\varphi^{*}$ is usc, satisfies $\varphi^{*}\leq\varphi^{*}\circ p_{\mathcal{X}}$, and $\varphi^{*}\circ i_{\mathcal{X}}=\varphi\circ i_{\mathcal{X}}$ is a uniform limit of restrictions to $\Delta_{\mathcal{X}}$ of $\theta$-psh functions, hence $\varphi^{*}$ is $\theta$-psh. ∎