The objects that the $(frac{1}{2}, frac{1}{2})$ representation of the Lorentz group act on

In the Physics from Symmetry book (pg. 80-83), the author uses the convention of dotted and undotted spinor indices such that the $(frac{1}{2},frac{1}{2})$ representation of the $SO(1,3)$ Lorentz group acts on $2times2$ Hermitian matrices denoted by $v^dot{b}_a$.

However, later on the author lowered the upper index $dot{b}$ to get $v_{adot{b}}$ and showed that transformations on the $2times2$ Hermitian matrices $v_{adot{b}}$ using the $(frac{1}{2}, frac{1}{2})$ representation are equivalent to Lorentz transformations.

The author then said that this shows that the $(frac{1}{2}, frac{1}{2})$ is the vector representation that we are used to when doing Lorentz transformations.

However, the $(frac{1}{2}, frac{1}{2})$ representation is supposed to act on $v^dot{b}_a$ instead of $v_{adot{b}}$. If the $(frac{1}{2}, frac{1}{2})$ representation acts on $v^dot{b}_a$, can we still show that it represents Lorentz transformations?