Let $\mathcal{X}^{\prime}\geq\mathcal{X}$ be two SNC models on which $\theta$ is determined. Then $p_{\mathcal{X}}\circ p_{\mathcal{X}^{\prime}}=p_{\mathcal{X}}$. By Proposition 7.5 (ii) this implies $\varphi\leq\varphi\circ p_{\mathcal{X}^{\prime}}\leq\varphi\circ p_{\mathcal{X}}\circ p_{\mathcal{X}^{\prime}}=\varphi\circ p_{\mathcal{X}}$, with equality on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$. Set $\tilde{\varphi}:=\lim_{\mathcal{X}}\varphi\circ p_{\mathcal{X}}$. Then $\tilde{\varphi}\geq\varphi$. On the other hand, it follows from Corollary 3.2 that $p_{\mathcal{X}}$ converges to the identity on $X$, so by upper semicontinuity of $\varphi$ we have $\varphi\geq\tilde{\varphi}$. ∎