On pdf page 39 (page 33 of the text) under Proposition 2.8, the following notation was used
$$U(\underline{V})=U(V_{i_0})\cdots U(V_{i_{m-1}})$$ $$U(\Gamma_m)=\{U(\underline{V}):V\in \Gamma_m\}$$
I double checked through the earlier parts of the book and never came across an explanation for the $U(V)$ notation. I thought at first the notation was to emphasize the choice of $U$ was a function of $V$ (i.e. "every $V\in \mathcal{V}$ lies in some $U(V)\in \mathcal{U}$").... but this just doesn't quite feel right.