It follows from Proposition 3.5 that $\varphi\circ\tau$ is a model function. To check that $\varphi\circ\tau$ is $\theta$-psh, we may assume $K$ algebraically closed as we have seen in the proof of Proposition 3.7. Moreover, we have seen that there is a strictly semistable model $\mathscr{X}^{\prime}$ of $X$ dominating $\mathscr{X}$ such that $f$ is determined on $\mathscr{X}^{\prime}$. Then

$\displaystyle\Delta\coloneqq S(\mathscr{X})\subset\Delta^{\prime}\coloneqq S(\mathscr{X}^{\prime})$

and $\varphi\circ\tau$ is constant along edges of $\Delta^{\prime}$ which are not contained in $\Delta$. By Proposition 3.5, there is a piecewise linear function $F^{\prime}$ on $\Delta^{\prime}$ with $\varphi=F^{\prime}\circ\tau^{\prime}$ for the canonical retraction $\tau^{\prime}\colon{X^{{\mathrm{an}}}}\to\Delta^{\prime}$ such that $mF^{\prime}$ is integral $\Gamma$-affine for a non-zero $m\in\mathbb{N}$. Moreover, the function $F^{\prime}$ is affine on the edges of $\Delta^{\prime}$. Let $F$ be the restriction of $F^{\prime}$ to $\Delta$. The same arguments as in [BFJ16a, Prop. 5.7] show that the $\theta$-psh function $\varphi$ is a uniform limit of functions of the form $\frac{1}{m}\log|\mathfrak{a}|$ with non-zero $m\in\mathbb{N}$ and with a vertical fractional ideal sheaf $\mathfrak{a}$ on $\mathscr{X}$. By [Ber99, Thm. 5.2(ii)], we deduce that $\varphi\leq\varphi\circ\tau$. Using the terminology introduced in Proposition 3.7, for all $x\in\Delta$ and $v\in T_{x}(\Delta^{\prime})$ we obtain $\lambda_{x,\nu}(F^{\prime})=\lambda_{x,\nu}(F)$ if $v\in T_{x}(\Delta^{\prime})$ and $\lambda_{x,\nu}(F^{\prime})\leq 0$ if $\nu\in T_{x}(\Delta^{\prime})\setminus T_{x}(\Delta)$. This implies

$\displaystyle 0$

$\displaystyle\leq\sum_{\nu\in T_{x}(\Delta^{\prime})}w_{x}(\nu)\lambda_{x,\nu}(F^{\prime})+\deg(\theta|_{\mathcal{C}_{x}})\leq\sum_{\nu\in T_{x}(\Delta)}w_{x}(\nu)\lambda_{x,\nu}(F)+\deg(\theta|_{\mathcal{C}_{x}})$

Here this first inequality comes from Proposition 3.7, since $\varphi$ is $\theta$-psh. Applying Proposition 3.7 again, we conclude that $\varphi\circ\tau=F\circ\tau$ is $\theta$-psh. ∎