I'm studying a graduate level course in QFT, and I have a potentially very stupid question that I can't find adressed anywhere (which I guess implies that I'm missing something fundamental). I'm currently working a problem where there is assumed to exist both a real scalar field $\Phi$ belonging to the adjoint representation of $SU(2)$ (an $SU(2)$ triplet), and a fermion field $\psi$ belonging to the fundamental representation (an $SU(2)$ doublet). As far as I understand, if we consider only the scalar, the Lagrangian for the theory should contain a kinetic term $$\frac{1}{2}D_\mu\Phi D^\mu\Phi,$$ where the covariant derivative should have the form $$D_\mu\phi_a = \partial_\mu\phi_a + g\varepsilon_{abc}A_\mu^b\phi_c,$$ (here $\Phi = \phi_a\tau_a$, with $\tau_a = \sigma_a/2$, and $g$ is the coupling strength).
However, since the fermion transforms in the fundamental representation, to make the theory gauge invariant, the Lagrangian should contain a term $$i\bar{\psi}\gamma^\mu D_\mu \psi,$$ but this time the covariant derivative should be $$D_\mu\psi = \partial_\mu\psi - igA_\mu^a\tau^a\psi,$$ clearly different from what we had before.
Now my question is, am I interpreting this correctly as there being two different covariant derivatives present, one for each different representation present in the theory, each with its own set of associated gauge fields $A_\mu^a$? Or should they be combined in some way? Or is something else going on entirely?