Chiral covariant derivative after $y^{\mu}=x^{\mu}+i\theta^{\dagger}\bar{\sigma}^{\mu}\theta$ coordinate transformation

Question

In formula (4.3.2) of this review on supersymmetry http://arxiv.org/pdf/hep-ph/9709356v6.pdf the chiral covariant derivative is given by

$$D_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}-i(\sigma^{\mu}\theta^{\dagger})_{\alpha}\partial_{\mu}$$

where $\theta,\theta^{\dagger}$ are the Grassmann coordinates and $\sigma^{\mu}=(I,\sigma^1,\sigma^2,\sigma^3)$. In section 4.4 when introducing the chiral superfield this coordinate transformation is considered

$$y^{\mu}=x^{\mu}+i\theta^{\dagger}\bar{\sigma}^{\mu}\theta$$

where $\bar{\sigma}^{\mu}=(I,-\sigma^1,-\sigma^2,-\sigma^3)$. In equation (4.4.5) it is claimed that this transformation makes the chiral covariant derivative be

$$D_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}-2i(\sigma^{\mu}\theta^{\dagger})_{\alpha}\frac{\partial}{\partial{}y^{\mu}}$$

can somebody prove this las equation?