Let $N_{1}$ and $N_{2}$ be lattices and $\Sigma_{i}$ a complete fan in $N_{i,\mathbb{R}}$, $i=1,2$. Let $H\colon N_{1}\to N_{2}$ be a linear map such that, for each $\sigma_{1}\in\Sigma_{1}$, there exists $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$. Let $p\in X_{\Sigma_{2},0}(K)$ and write $A\colon N_{1,\mathbb{R}}\to N_{2,\mathbb{R}}$ for the affine map $A=H+{\operatorname{val}}(p)$. Let $\|\cdot\|$ be an approachable toric metric on ${\mathcal{O}}(D_{\Psi_{2}})^{{\text{\rm an}}}$. Then

$$\psi_{\varphi_{p.H}^{\ast}\|\cdot\|}=A^{*}\psi_{\|\cdot\|}.$$

Moreover, the Legendre-Fenchel dual of this function is given by

$$\psi_{\varphi_{p.H}^{\ast}\|\cdot\|}^{\vee}=(H^{\vee})_{*}\big(\psi^{\vee}_{\|\cdot\|}-{\operatorname{val}}(p)\big).$$