Derivation of fermionic partition function, how does commutation work?

When deriving the fermionic partition function with coherent states $|psirangle$ we make the following step

$$
mathcal Z=int d(barpsi,psi) e^{-sum_ibarpsi_i psi_i}sum_nlangle n|psiranglelanglepsi|e^{-beta(hat H – muhat N)}|nrangletag{1}
$$

$$
mathcal Z=int d(barpsi,psi) e^{-sum_ibarpsi_i psi_i}sum_nlangle zetapsi|e^{-beta(hat H – muhat N)}|nranglelangle n|psirangletag{2}
$$

Why does this work, why can we just commute the brakets and where does the $zeta=-1$ come from?