We investigate the infinite version of the $k$-switch problem of Greenwell and Lovász. Given infinite cardinals $κ$ and $λ$, for functions $x,y\in {}^λκ$ we say that they are totally different if $x(i)\ne y(i)$ for each $i\in λ$. A function $F:{}^λκ\longrightarrow κ $ is a proper coloring if $F(x)\ne F(y)$ whenever $x$ and $y$ are totally different elements of ${}^λκ $. We say that $F$ is weakly uniform iff there are pairwise totally different functions $\{r_α:α<κ\}\subset {}^λκ$ such that $F(r_α)=α$; $F$ is tight if there is no proper coloring $G:{}^λκ\longrightarrow κ$ such that there is exactly one $x\in {}^λκ$ with $G(x)\ne F(x)$. We show that given a proper coloring $F:{}^λκ\to κ$, the following statements are equivalent $F$ is weakly uniform, there is a $κ ^{+}$-complete ultrafilter $\mathscr{U}$ on $λ$ and there is a permutation $π\in Symm(κ)$ such that for each $x\in {}^λκ$ we have $$F(x)=π(α)\ \Longleftrightarrow \ \{i\in λ: x(i)=α\} \in \mathscr{U}.$$ We also show that there are tight proper colorings which cannot be obtained such a way.