Let $A\colon Q_{\mathbb{R}}\to N_{\mathbb{R}}$ be an affine map defined as $A=H+u_{0}$ for a linear map $H$ and a point $u_{0}\in N_{\mathbb{R}}$. Let $f\in\mathscr{P}_{N_{\mathbb{R}}}$ (respectively $f\in{\overline{\mathscr{P}}}_{N_{\mathbb{R}}}$) with ${\operatorname{dom}}(f)\cap\operatorname{im}(A)\not=\emptyset$ and $g\in\mathscr{P}_{Q_{\mathbb{R}}}$ (respectively $g\in{\overline{\mathscr{P}}}_{Q_{\mathbb{R}}}$) such that $\operatorname{stab}(g)\cap\operatorname{im}(H^{\vee})\not=\emptyset$. Then $A^{\ast}f\in{\mathscr{P}}_{Q_{\mathbb{R}}}$ (respectively $A^{*}f\in{\overline{\mathscr{P}}}_{Q_{\mathbb{R}}}$) and $A_{*}g\in\mathscr{P}_{N_{\mathbb{R}}}$ (respectively $A_{*}g\in{\overline{\mathscr{P}}}_{N_{\mathbb{R}}}$). Moreover,

1. (1)

   $\operatorname{stab}(A^{\ast}f)=H^{\vee}(\operatorname{stab}(f))$, $(A^{\ast}f)^{\vee}=(H^{\vee})_{\ast}(f^{\vee}-u_{0})$ and, for all $y\in\operatorname{stab}(A^{\ast}f)$,

   $$(A^{\ast}f)^{\vee}(y)=\max_{x\in(H^{\vee})^{-1}(y)}(f^{\vee}(x)-\langle x,u_{0}\rangle);$$

2. (2)

   $\operatorname{stab}(A_{\ast}g)=(H^{\vee})^{-1}(\operatorname{stab}(g))$, $(A_{\ast}g)^{\vee}=(H^{\vee})^{\ast}(g^{\vee})+u_{0}$ and, for all $u\in{\operatorname{dom}}(A_{*}g)$,

   $$A_{*}g(u)=\max_{v\in A^{-1}(u)}g(v).$$