# Why does $A_\mu$ undergo an adjoint representation matrix transformation?

This question pertains to the following passage from Weinberg's second volume on QFT. It appears on page 4, section 15.1.

To make the Lagrangian invariant, we need a field $$A^\alpha_\mu$$, whose transformation rule involves a term $$\partial_\mu \epsilon^\mu$$, which can be used to cancel the second term in Eq. (15.1.8). Since this field carries an $$\alpha$$-index, we would expect it also undergo a matrix transformation like Eq. (15.1.1), but with $$t_\alpha$$ replaced with the adjoint representation matrices (15.1.6).

The bold emphasis is mine, because it is this expectation that I'm not grasping. First though, here are the referenced equations,

$$\delta\left(\partial_{\mu} \psi_{\ell}(x)\right)=i \epsilon^{\alpha}(x)\left(t_{\alpha}\right)_{\ell}^{m}\left(\partial_{\mu} \psi_{m}(x)\right)+i\left(\partial_{\mu} \epsilon^{\alpha}(x)\right)\left(t_{\alpha}\right)_{\ell}^{m} \psi_{m}(x) \tag{15.1.8}$$

This passage is motivating the need for a covariant derivative, since Eq. (15.1.8) shows that derivative terms of the matter fields, $$\partial_\mu\psi_l(x)$$ do not transform in the same way as the fields themselves: $$\delta \psi_{\ell}(x)=i \epsilon^{\alpha}(x)\left(t_{\alpha}\right)_{\ell}^{m} \psi_{m}(x) \tag{15.1.1}$$

and lastly, the adjoint representation matrices are those defined by the structure constants themselves,

$$\left(t_{\alpha}^{\mathrm{A}}\right)^{\beta}{ }_{\gamma} \equiv-i C_{\gamma \alpha}^{\beta} \tag{15.1.6}$$

Now to my confusion. When Weinberg says:

Since this field carries an $$\alpha$$-index, we would expect it also undergo a matrix transformation like Eq. (15.1.1)...

Even this is surprising to me, because $$\psi_l$$ doesn't carry an $$\alpha$$-index. It's not obvious to me why $$A^\alpha_\mu$$ carrying this index should mean necessarily that it will undergo a matrix transformation.Furthermore, it's entirely unclear to me why the $$t_\alpha$$ should be replaced by the adjoint representation matrices. Is that something we should also expect from looking at how $$\partial_{\mu} \psi_{l}(x)$$ transforms?