  We establish the direct $d=2$ on-shell bosonization
$\psi_{L}(x_{+})=e^{i\xi(x_{+})}$ and
$\psi_{R}^{\dagger}(x_{-})=e^{i\xi(x_{-})}$ in path integral formulation by
deriving the off-shell relations
$\psi_{L}(x)\psi_{R}^{\dagger}(x)=\exp[i\xi(x)]$ and
$\psi_{R}(x)\psi_{L}^{\dagger}(x)=\exp[-i\xi(x)]$. Similarly, the on-shell
bosonization of the bosonic commuting spinor,
$\phi_{L}(x_{+})=ie^{-i\xi(x_{+})}\partial^{+}e^{-i\chi(x_{+})}$,
$\phi^{\dagger}_{R}(x_{-})=e^{-i\xi(x_{-})-i\chi(x_{-})}$ and
$\phi_{R}(x_{-})=ie^{i\xi(x_{-})}\partial^{-}e^{+i\chi(x_{-})}$,
$\phi^{\dagger}_{L}(x_{+})=e^{i\xi(x_{+})+i\chi(x_{+})}$, is established in
path integral formulation by deriving the off-shell relations
$\phi_{L}(x)\phi^{\dagger}_{R}(x)=ie^{-i\xi(x)}\partial^{+}e^{-i\chi(x)}$ and
$\phi_{R}(x)\phi^{\dagger}_{L}(x)=ie^{i\xi(x)}\partial^{-}e^{i\chi(x)}$.
