= Antilinear charge-conjugation intertwiner
{title2=$\psi^c=B\psi^*,\qquad\gamma^\mu B=-B\gamma^{\mu*}$}
In the complex-conjugate convention for <charge conjugation of a Dirac field>, a unitary matrix $B$ obeys the displayed <gamma matrix> identity. For $M^{\rho\sigma}=\frac14[\gamma^\rho,\gamma^\sigma]$, two minus signs cancel to give $M^{\rho\sigma}B=BM^{\rho\sigma*}$. Real <Lorentz transformation> parameters therefore give $S B=B S^*$, so $B\psi^*$ has the same <Spinor representation of the Lorentz group> transformation law as $\psi$. Complex conjugating the free <Dirac equation> and multiplying by $B$ proves that the conjugate also solves it for real mass. This $B$ is the matrix multiplying $\psi^*$; the convention multiplying $\bar\psi^T$ instead uses $B(\gamma^0)^T$.
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