= Solution
Writing the source interaction as $\operatorname{Tr}(J^\mu A_\mu)$, gauge invariance requires the matrix current to transform in the <Adjoint representation>,
$$
\boxed{J^\mu\mapsto UJ^\mu U^\dagger}.
$$
The fermion bilinear with this transformation law is
$$
\boxed{J^{\mu a}=g\,\bar\psi_i\gamma^\mu(T^a)_{ij}\psi_j},
\qquad
J^\mu=J^{\mu a}T_a,
$$
up to a convention-dependent overall coupling or generator normalization. It is the <Noether current> that appears when the free derivative is replaced by $D_\mu=\partial_\mu-igA_\mu$.
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