Let $[n]=\{0,1\}^n$ equipped with the product order. I consider the small category $\widehat{\square}$ of the category of posets generated by the coface maps $\delta^\alpha_i:[n-1]\to [n]$ defined by $\delta^\alpha_i:(\epsilon_1,\dots,\epsilon_{n-1}) \mapsto (\epsilon_1,\epsilon_{i-1},\alpha,\epsilon_i,\dots,\epsilon_{n-1})$ and by the strictly increasing maps $f:[n]\to [n]$.

The small category $\widehat{\square}$ contains the symmetry maps (the ones permuting the coordinates).

Informally, I would like to remove the symmetry maps from $\widehat{\square}$, and only them, to obtain a subcategory of $\widehat{\square}$. Every strictly increasing map $f=(f_1,\dots,f_n):[n]\to [n]$ gives rise to another strictly increasing map by permuting the coordinates. I need to find a way to make a choice among all permutations.

Every strictly increasing map $f=(f_1,\dots,f_n):[n]\to [n]$ satisfies the equalities $f_i(x_1,\dots,x_n) = \max_{(\epsilon_1,\dots,\epsilon_n)\in f_i^{-1}(1)} \min \{x_k\mid \epsilon_k=1\}$ for all $1\leq i\leq n$ (see https://mathoverflow.net/a/429941/24563).

Question: In the formula above, is there a way to put a total order on the coordinates by using the syntax of the formula ?

The motivation is that I work with the presheaves on $\widehat{\square}$ that I call transverse sets. They are a generalization of the category of precubical sets adapted for studying the directed homotopy of concurrency. And I would like to define the non-symmetric transverse sets.