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 $f$ be a concave function on $N_{\mathbb{R}}$ such that ${\operatorname{dom}}(f)\cap\operatorname{im}(A)\not=\emptyset$ and $g$ a concave function on $Q_{\mathbb{R}}$ such that $\operatorname{stab}(g)\cap\operatorname{im}(H^{\vee})\not=\emptyset$. Then

1. (1)

   $\operatorname{stab}(A_{\ast}g)=(H^{\vee})^{-1}(\operatorname{stab}(g))$ and

   $$(A_{\ast}g)^{\vee}=(H^{\vee})^{\ast}(g^{\vee})+u_{0};$$

2. (2)

   $H^{\vee}(\operatorname{stab}(f))\subset\operatorname{stab}(A^{\ast}f)\subset{\overline{H^{\vee}(\operatorname{stab}(f))}}$ and

   $$(A^{\ast}{\operatorname{cl}}(f))^{\vee}={\operatorname{cl}}((H^{\vee})_{\ast}(f^{\vee}-u_{0}));$$

3. (3)

   if $\operatorname{ri}({\operatorname{dom}}(f))\cap\operatorname{im}(A)\not=\emptyset$ then $\operatorname{stab}(A^{\ast}f)=H^{\vee}(\operatorname{stab}(f))$ and, for all $y$ in this set,

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

   Moreover, for $y\in\operatorname{ri}(\operatorname{stab}(A^{\ast}f))$, a point $x\in(H^{\vee})^{-1}(y)$ realizes this maximum if and only if $x\in\partial f(Av)$ for a $v\in Q_{\mathbb{R}}$ such that $y\in\partial(A^{*}f)(v)$.