The non-homogeneous part of the Yang-Mills equations is given by
$$D\star F=\star J,$$
where $D=d+A$ is the covariant derivative, $\star$ is the Hodge star and $J$ is the source current.
Under a gauge transformation $U(x)$, the above equation should stay invariant. Under a gauge transformation $U(x)$, $D\star F$ is transformed to
$$D^{\prime}\star F^{\prime},$$
where $F^{\prime}=UFU^{-1}$. One finds
$$D^{\prime}\star F^{\prime}=U(D\star F)U^{-1}.$$
Thus, if the equation of motion is invariant under the gauge transformation, one expects that the current $J$ should transform as
$$J^{\prime}=UJU^{-1}.$$
However, take the spinor-QCD as an example, the gauge current is given by
$$J^{a}_{\mu}(x)=\bar{\Psi}\gamma_{\mu}T^{a}\Psi,$$
where $T_{a}$ is the generator of the gauge group. Under a gauge transformation, the spinor transforms as
$$\Psi^{\prime}(x)=U(x)\Psi(x).$$
It seems to me that this spinor current doesn't transform as expected.
What is wrong?