= Solution
Write $\psi^C=\mathcal C\bar\psi^T$ and $\bar\psi^C=\psi^T\mathcal C$. Anticommuting the spinor fields and using $\mathcal C^{-1}(\gamma^\mu)^T\mathcal C=-\gamma^\mu$ gives
$$
\bar\psi^C\psi^C=\bar\psi\psi,
\qquad
\bar\psi^C\gamma^\mu\psi^C=-\bar\psi\gamma^\mu\psi.
$$
The free Dirac terms are invariant up to the total derivative used to move the derivative between the two anticommuting fields. The QED interaction $-e\bar\psi\gamma^\mu\psi A_\mu$ is also invariant because both the <Dirac current> and $A_\mu$ are odd. Together with $F_{\mu\nu}F^{\mu\nu}$ invariance, this proves charge-conjugation invariance of <quantum electrodynamics>.
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