How is it a single variable optimization problem?

Question

In example 5.4 of Convex Optimization book (by Vandenberghe and Boyd) they show that the optimization problem $$\min. f_0(x)=\sum_{i=1}^nf_i(x_i),\\ \text{s.t. }a^Tx=b$$ has a dual problem $$\max. ~-bv-\sum_{i=1}^nf_i^*(-va_i)$$ with scalar variable $v$. Next, they say that the dual problem is a single variable problem. I do not understand how is it a single variable problem. Because in order to get $f_i^*(-va_i)$ we have to maximize over individual $x_i$'s. Where am I wrong in this understanding? Any help in this regard will be much appreciated. Thanks in advance.