= Operator conjugation of a chiral field component
{title2=$\hat C\psi_L\hat C^{-1}=P_L\psi^c=(\psi_R)^c$}
The unitary charge-conjugation operator acts on field operators and commutes with numerical <chiral projectors>, so $\hat C(P_L\psi)\hat C^{-1}=P_L\psi^c$. This component is left-chiral. The algebraic conjugate of an already projected spinor instead satisfies $(P_L\psi)^c=P_R\psi^c$, because its adjoint carries the opposite projector. Keeping the two operations separate avoids a false chirality conclusion.
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