  For all positive integers $n$ and all homotopy modules $M_*$, we define
certain operations $\underline{\operatorname{K}}^{\operatorname{MW}}_n
\rightarrow M_*$ and show that these generate the $M_*(k)$-module of all (in
general non-additive) operations
$\underline{\operatorname{K}}^{\operatorname{MW}}_n \rightarrow M_*$ in a
suitable sense, if $M_*$ is $\mathbb{N}$-graded and has a ring structure. This
also allows us to explicitly compute the abelian group
$\operatorname{Op}(\underline{\operatorname{K}}^{\operatorname{MW}}_n,\underline{\operatorname{K}}^{\operatorname{MW}}_m)$
and all operations between related theories such as Milnor, Witt and
Milnor-Witt K-theory.
