Let $H\colon Q_{\mathbb{R}}\to N_{\mathbb{R}}$ be a linear map, $u_{0}\in N_{\mathbb{R}}$ and $A=H+u_{0}$ the associated affine map. Let $f$ be a concave function on $N_{\mathbb{R}}$, then

1. (1)

   $\partial(A^{*}f)(v)\supset H^{\vee}\partial f(Av)$ for all $v\in Q_{\mathbb{R}}$;

2. (2)

   if either $\operatorname{ri}({\operatorname{dom}}(f))\cap\operatorname{im}(A)\neq\emptyset$ or $f$ is piecewise affine and ${\operatorname{dom}}(f)\cap\operatorname{im}(A)\neq\emptyset$, then for all $v\in Q_{\mathbb{R}}$ we have

   $$\partial(A^{*}f)(v)=H^{\vee}\partial f(Av).$$