Question with computing the chiral scalar condensate in lattice QCD about the dagger of the discretized fermionic propagator

My professor defined, given $a$ the lattice spacing the massive propagator according to the Wilson prescription as:
$$D_m equiv D+m left( 1-frac{aD}{2} right)$$
Then he computes the chrial condensate $S_0=bar{psi} psi$ as:
$$frac{left< S_0 right>}{N_f} = left< Tr left( 1-frac{aD}{2} right)frac{1}{D_m} right>$$
Now up to this point I agree with him but here comes the issues: he says that
$$D_mD_m^dagger = DD^dagger + m^2left( 1-frac{a^2DD^dagger}{2^2} right)$$
while I would compute $D_m^dagger=D^dagger+mleft( 1-frac{aD^dagger}{2} right)$ which would give a much more complex expression for $D_mD_m^dagger$. So what am I doing wrong with this dagger operator?