I am reading this paper about constructive renormalization for fermions and I got a really basic question about it. There, the effective Lagrangian (with UV cutoff $\Lambda_{0}$ and IR cutoff $\Lambda$) is given by the expression: $$L^{\text{eff}}(\Lambda) = \langle \xi, C^{\Lambda}\xi\rangle -[\lim_{\Lambda_{0}\to \infty}(\ln Z_{V}^{\Lambda,\Lambda_{0}}(\xi)-\langle \xi, C_{\Lambda}^{\Lambda_{0}}\xi\rangle)] \tag{1}\label{1}$$ where $C = C_{\Lambda}^{\Lambda_{0}}$ is the regularized covariance (here a matrix) and $\xi$ are fermionic Grassmann variables. The paper then tells us that the coefficients (for the expansion in the fields) of this effective Lagrangian are given by two-point functions $\Gamma_{2p}^{\Lambda}(\{y\},\{z\}) = \lim_{\Lambda_{0}\to \infty}\Gamma_{2p}^{\Lambda\Lambda_{0}}(\{y\},\{z\})$, with: $$\Gamma_{2p}^{\Lambda\Lambda_{0}}(\{y\},\{z\}) = \Gamma_{2p}^{\Lambda\Lambda_{0}}(y_{1},...,y_{p},z_{1},...,z_{p}) = \lim_{V\to \infty}\frac{\delta^{2p}}{\delta\xi(z_{1})\cdots \delta\xi(z_{p})\delta\bar{\xi}(y_{1})\cdots \delta\bar{\xi}(y_{p})}(\ln Z_{V}^{\Lambda\Lambda_{0}}-\langle \xi,C_{\Lambda}^{\Lambda_{0}}\xi\rangle)(C_{\Lambda}^{\Lambda_{0}})^{-1}(\xi)\bigg{|}_{\xi=0}. \tag{2}\label{2}$$
So, my question is: where does expression (\ref{2}) come from? I mean, the coefficients of the expansion shouldn't be just: $$\lim_{V\to \infty}\frac{\delta^{2p}}{\delta\xi(z_{1})\cdots \delta\xi(z_{p})\delta\bar{\xi}(y_{1})\cdots \delta\bar{\xi}(y_{p})}L^{\text{eff}}(\Lambda)\bigg{|}_{\xi=0}$$ instead of (\ref{2})? How to obtain (\ref{2}) instead? Where does this $(C_{\Lambda}^{\Lambda_{0}})^{-1}(\xi)$ even come from?
