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 the inverse image of $f$ by $A$ is defined as

$$A^{\ast}f\colon Q_{\mathbb{R}}\longrightarrow\mathbb{R},\quad v\longmapsto f\circ A(v),$$

and the direct image of $g$ by $A$ is defined as

$$A_{\ast}g\colon N_{\mathbb{R}}\longrightarrow\mathbb{R},\quad u\longmapsto\sup_{v\in A^{-1}(u)}g(v).$$