= Charge conjugation of the Dirac adjoint
{title2=$\overline{\psi^c}=-\psi^TC^{-1}$}
For $\psi^c=C\bar\psi^T$, use a unitary <charge-conjugation matrix> with $C^{-1}\gamma^0C=-(\gamma^0)^T$ and Hermitian $\gamma^0$ of square one. The <Dirac adjoint> then satisfies
$$
\overline{\psi^c}=\psi^T(\gamma^0)^*C^{-1}\gamma^0=-\psi^TC^{-1}.
$$
If the unitary <charge conjugation> operator sends $\psi$ to $\eta_C\psi^c$, it sends $\bar\psi$ to $-\eta_C^*\psi^TC^{-1}$. Antisymmetry $C^T=-C$ also gives $(\psi^c)^c=\psi$. The unitary operator acting on fields is not an <antiunitary operator> acting on numerical coefficients.
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