Let $C\subseteq Z$ be the curve associated with the cone $\sigma\cap\sigma^{\prime}$. We have

$$\begin{cases}W^{\sigma^{\prime}}_{i}=W^{\sigma}_{i}-(C\cdot D_{i})W^{\sigma}_{i_{0}}&\textrm{ \hskip 10.22217ptfor }i\in L_{\sigma\sigma^{\prime}}\\ W^{\sigma^{\prime}}_{i_{\infty}}=-W^{\sigma}_{i_{0}}&\\ W^{\sigma^{\prime}}_{j}=W^{\sigma}_{j}-(C\cdot D_{j})W^{\sigma}_{i_{0}}&\textrm{ \hskip 10.22217ptfor }j\in J\end{cases}$$

In other words, the relation $W^{\sigma^{\prime}}=(M_{\mathcal{B}^{\prime}\mathcal{B}}\otimes\Id)W^{\sigma}$ holds.