On P&S's QFT page 302, eq.(9.73) defined the generating functional for the Dirac field. $$Z[\bar{\eta}, \eta]=\int \mathcal{D} \bar{\psi} \mathcal{D} \psi \exp \left[i \int d^4 x[\bar{\psi}(i \not \partial-m) \psi+\bar{\eta} \psi+\bar{\psi} \eta]\right] \tag{9.73}$$ Then we need to shift $\psi$ to complete the square, here is my attempt: $$\psi^{\prime}\equiv \psi+(i\not\partial-m)^{-1}\eta \tag{A}$$ I am troubled for $\overline{\psi}$, according to (A): $$\begin{aligned} \overline{\psi}^{\prime}&=\overline{\psi}+ \eta^{\dagger}\left[(i\not\partial-m)^{-1}\right]^{\dagger}\gamma^0 \\ &=\overline{\psi}+ \overline{\eta}\gamma^0\left[(i\not\partial-m)^{-1}\right]^{\dagger}\gamma^0 \\ &=\overline{\psi}+ \overline{\eta}\gamma^0\left[(i\not\partial-m)^{\dagger}\right]^{-1}\gamma^0 \\ &=\overline{\psi}+ \overline{\eta}\left[\gamma^0(i\not\partial-m)^{\dagger}\gamma^0\right]^{-1} \\ &=\overline{\psi}+ \overline{\eta}\left[\gamma^0(-i\partial_{\mu}(\gamma^{\mu})^{\dagger}-m)\gamma^0\right]^{-1} \\ &=\overline{\psi}+ \overline{\eta}\left[(-i\partial_{\mu}\gamma^{\mu}-m)\right]^{-1} \\ &=\overline{\psi}- \overline{\eta}\left[(i\not\partial+m)\right]^{-1} \\ \end{aligned} $$ where I used $\gamma^0 (\gamma^{\mu})^{\dagger} \gamma^0=\gamma^{\mu}$. But this last look quite wrong! Since we have $(i\not\partial+m)^{-1}$. So where is my problem?
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