Application of waist inequality to entropy and mean dimension: II

Let $X$ be a full-shift on the alphabet $[0,1{]}^a$ and let $(Y,S)$ be an arbitrary dynamical system. We prove that any equivariant continuous map from $X$ to $Y$ has conditional metric mean dimension not less than $a-\mathrm{m}\mathrm{d}\mathrm{i}\mathrm{m}(Y,S)$. This solves a problem posed in a paper of Shi–Tsukamoto (2023).