The naive Milnor–Witt K-theory relations in the stable motivic homotopy groups over a base

We construct the homomorphism of presheaves ${\mathrm{K}}^\mathrm{MW}_* \to {\pi}^{*,*}$ over an arbitrary base scheme $S$, where $\mathrm{K}^\mathrm{MW}$ is the (naive) Milnor-Witt K-theory presheave. Also we discuss some partly alternative proof (or proofs) of the isomorphism of sheaves $\unKMW_n\simeq \underline{\pi}^{n,n}_s$, $n\in \mathbb Z$, over a filed $k$ originally proved in \cite{M02} and \cite{M-A1Top}.