Is there a non-constant continous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and matrices $A=\begin{pmatrix} a_1 & 0\\ 0 & a_2\\ \end{pmatrix}$ and $B=\begin{pmatrix} b_1 & 0\\ 0 & b_2\\ \end{pmatrix}$ for which there does not exist any matrices $C\in \mathrm{Mat}_{d\times 2},D \in \mathrm{Mat}_{2\times d}$ and vectors $c \in \mathbb{R}^d,e\in \mathbb{R}^2$ such that:

$ D f_d\left(C \begin{pmatrix} x\\ x \end{pmatrix} +c \right) + e = Af_2 \left(B \begin{pmatrix} x\\ x \end{pmatrix} \right) \qquad (\forall x \in \mathbb{R}) $

$D,e,C,$ and $c$ only have non-zero entries,

where $f_i(x)= (f(x_1),\dots,f(x_n))$.

1more comment