When I am asked to evaluate, $\mathbf{U^{\alpha}_{~,~~\beta}}$ for all $\alpha$ and $\beta$, what does it mean ? I have not able to understand this notation.

In case of $\mathbf{g(~~,~\bar{A})}$ I understand that the blank in the metric tensor means it acts like a one-form which takes in a vector and outputs a real number. For this case it makes sense to me because it is a

$\begin{bmatrix} 0 \\ 2 \end{bmatrix}$ tensor. How do we understand the same for a mixed tensor like the one above ?

Reference : Chapter 3, Problem 30 : A First Course in General Relativity, Second Edition, B. Schutz, pg 82