= Solution
The <Dirac adjoint> is $\bar u=u^\dagger\gamma^0$. Because $\gamma^5$ is <Hermitian> and anticommutes with $\gamma^0$,
$$
\overline{(1-\gamma^5)u_A}=u_A^\dagger(1-\gamma^5)\gamma^0=\bar u_A(1+\gamma^5).
$$
Anticommutation with $\gamma^\mu$ then gives
$$
\boxed{\overline{(1-\gamma^5)u_A}\gamma^\mu u_B=\bar u_A(1+\gamma^5)\gamma^\mu u_B=\bar u_A\gamma^\mu(1-\gamma^5)u_B}.
$$
Thus the left-chiral <charged current> can be written with its <chiral projector> acting on either external <Dirac spinor>. More precisely, $\bar u_A\gamma^\mu P_Lu_B=\overline{P_Lu_A}\gamma^\mu P_Lu_B$, since the opposite projection vanishes. The <Dirac adjoint> reverses the matrix projector, $\overline{P_Lu}=\bar uP_R$; overlooking this reversal would give a wrong zero. The source uses $1-\gamma^5=2P_L$, so no additional factor of two is introduced when moving that entire factor.
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