Verify (12.125) using (5.39).
With the replacements (12.123), the first charged current of (12.121) becomes
$$
\begin{aligned}
\left(J_{c a}^\mu\right)_C & =U_{c a}\left(\bar{u}_c\right)_C \gamma^\mu\left(1-\gamma^5\right)\left(u_a\right)_C \\
& =-U_{c a} u_c^T C^{-1} \gamma^\mu\left(1-\gamma^5\right) C \bar{u}_a^T \\
& =U_{c a} u_c^T\left[\gamma^\mu\left(1+\gamma^5\right)\right]^T \bar{u}_a^T \\
& =(-) U_{c a} \bar{u}_a \gamma^\mu\left(1+\gamma^5\right) u_c .
\end{aligned}
$$
The above procedure is exactly analogous to that used to obtain the charge-conjugate electromagnetic current, (5.40).
The parity operation $P=\gamma^0$, see (5.62), and so
$$
P^{-1} \gamma^\mu\left(1+\gamma^5\right) P=\gamma^{\mu \dagger}\left(1-\gamma^5\right),
$$
see (5.9)-(5.11). Thus,
$$
\left(J_{c a}^\mu\right)_{C P}=(-) U_{c a} \bar{u}_a \gamma^{\mu \dagger}\left(1-\gamma^5\right) u_c,
$$
and hence
$$
\Re_{C P}-U_{c a} U_{d b}^*\left[\bar{u}_a \gamma^\mu\left(1-\gamma^5\right) u_c\right]\left[\bar{u}_b \gamma_\mu\left(1-\gamma^5\right) u_d\right] .
$$
We can now compare $9 \pi_{C P}$ with $9 \pi^{\dagger}$ of (12.122). Provided the elements of the matrix $U$ are real, we find
$$
\Re_{C P}=\Re^{\dagger} \text {, }
$$
and the theory is $C P$ invariant. At the four-quark (u,d,c,s) level, this is the case, as the $2 \times 2$ matrix $U,(12.106)$, is indeed real. However, with the advent of the $\mathrm{b}$ (and t) quarks, the matrix $U$ becomes the $3 \times 3$ Kobayashi-Maskawa (KM) matrix. It now contains a complex phase factor $e^{i \delta}$. Then, in general, we have
$$
\pi_{C P} \neq \mathscr{R}^{\dagger} \text {, }
$$
and the theory necessarily violates $C P$ invariance.
In fact, a tiny $C P$ violation had been established many years before the introduction of the KM matrix. The violation was discovered by observing the decays of neutral kaons. These particles offer a unique "window" through which to look for small $C P$ violating effects. We discuss this next.