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It is not significant how to get to the relations between $\sigma$ and $S$. You can think about the equalities for $\sigma^z$ and $\sigma^x$ as an operator change of variables that someone came up with, perhaps Jordan and Wigner. What is important, this change of variables is consistent with Pauly's matrices properties. Relations $$ \sigma_j^y = i\sigma_j^x\sigma_j^z = -i\sigma_j^z\sigma_j^x, $$ $$ S_j^y = iS_j^xS_j^z = -iS_j^zS_j^x,\quad (S_j^x)^2 = 1,\quad S_j^\alpha S_k^\beta = S_k^\beta S_j^\alpha, \quad\forall\ j\neq k,\ \alpha, \beta $$ together with the equalities for $\sigma^x$ and $\sigma^z$ lead to the following equality for $\sigma^y$: $$ \sigma_j^y = -\prod_{k<j} S_k^x\ S_j^y S_{j+1}^z $$ It is straightforward to check the validity of all the usual Pauly's matrices relations now. It is also easy to obtain formulas for $\sigma_j^\alpha\sigma_{j+1}^\alpha$ for $\alpha = x, y, z$ and to find hamiltonian dual to that of the Heisenberg model.