The relation in Eq. 2.1.7 boils down to the two following:

$$\begin{cases}\sum_{l\in L}(C\cdot D_{l})u_{l}=0\\ (C\cdot D_{j})+\sum_{l\in L}\lambda_{j,l}(C\cdot D_{l})=0.\\ \end{cases}$$

The first one follows directly from Eq. 1.2.4 in the fan $\Sigma$ of $Z$; the second comes from the construction of $\lambda_{l}$, and in particular from $C\cdot D_{j}=C\cdot F_{j}=-C\cdot\sum_{l\in L}\lambda_{j,l}Z_{l}$. ∎