There is a contradiction in the Johnson’s algorithm presented in CLRS edition 3 page 700 that I can't understand.
Johnson’s algorithm uses the technique of reweighting, which works as follows: for each edge $(u,v)$ that may has a negative weight, a new weight is assigned which is as follow:
Given a weighted, directed graph $G=(V,E)$ with weight function $w:E \rightarrow \mathbb R$, let $h:V \rightarrow \mathbb R$ be any function mapping vertices to real numbers. For each edge $(u, v) \in E$, define $\hat w(u,v) = w(u,v) + h(u) - h(v)$
The book in page 702 proves that $G$ has a negative-weight cycle using weight function $w$ if and only if $G$ has a negative-weight cycle using weight function $\hat w$.
On the other hand in page 702 it shows that for each $(u,v) \in E, \hat w(u,v) \geq 0$ .
Here is a my problem, how a negative-weight cycle of $G$ using $w$ weight function produces a negative-weight cycle using $\hat w$ weight function when it already has been proved that $\hat w(u,v) \geq 0$ for every edge $(u,v)$?