= Solution
For a <normal ordering> of a <fermion bilinear>, the result of part (a) and the <fermionic sign> from interchanging the fields give
$$
\hat C(\bar\psi\Gamma\psi)\hat C^{-1}
=-\psi^T C^{-1}\Gamma C\bar\psi^T
=\bar\psi(C^{-1}\Gamma C)^T\psi.
$$
For the <Dirac current>, $(C^{-1}\gamma^\mu C)^T=-\gamma^\mu$. For the <axial current>, the definition of $\gamma^5$ and the <gamma matrix> identities imply
$$
C^{-1}\gamma^5C=\gamma^{5T},\qquad
(C^{-1}\gamma^\mu\gamma^5C)^T
=-\gamma^5\gamma^\mu=\gamma^\mu\gamma^5.
$$
Thus the <charge conjugation of fermion bilinears> gives
$$
\boxed{\hat C(\bar\psi\gamma^\mu\psi)\hat C^{-1}=-\bar\psi\gamma^\mu\psi,\qquad
\hat C(\bar\psi\gamma^\mu\gamma^5\psi)\hat C^{-1}=+\bar\psi\gamma^\mu\gamma^5\psi.}
$$
The <Dirac current> is \b[odd under <charge conjugation>], and the <axial current> is \b[even under <charge conjugation>]. The <normal ordering> specifies the usual treatment of coincident-field contact terms.
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