 Solution

The Dirac adjoint is $\bar{u}=u^{†}{\gamma }^0$. Because ${\gamma }^5$ is Hermitian and anticommutes with ${\gamma }^0$,Anticommutation with ${\gamma }^{\mu }$ then givesThus the left-chiral charged current can be written with its chiral projector acting on either external Dirac spinor. More precisely, ${\bar{u}}_A{\gamma }^{\mu }{P}_L{u}_B=\overline{P_L{u}_A}{\gamma }^{\mu }{P}_L{u}_B$, since the opposite projection vanishes. The Dirac adjoint reverses the matrix projector, $\overline{P_{L}u}=\bar{u}{P}_R$; 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.