We need to compute the parallel transport of the vectors

$$(u_{0},\ldots,u_{n-1})\coloneqq(v_{D_{i_{0}}}-v_{D_{i_{n}}},v_{D_{i_{1}}}-v_{D_{i_{n}}},\ldots,v_{D_{i_{n-1}}}-v_{D_{i_{n}}})$$

along the loop $\gamma$. By Proposition 3.1.1 the $\mathbb{Z}$-affine structure on $\Star(\tau_{C})$ induced by $\rho_{\mathscr{X}}$ is described by the chart which has the following vertices:

$v_{0}=(1,0,\ldots,0)$, $v_{1}=(0,1,\ldots,0),\,\ldots,\,v_{n}=0$ and $v_{\infty}=(-1,b_{i_{1}},\ldots,b_{i_{n-1}});$

while the $\mathbb{Z}$-affine structure induced by $\rho_{\mathscr{X}^{\prime}}$ is given by:

$v^{\prime}_{0}=(1,0,\ldots,0)$, $v^{\prime}_{1}=(0,1,\ldots,0),\,\ldots,\,v^{\prime}_{n}=0$ and $v^{\prime}_{\infty}=(-1,b^{\prime}_{i_{1}},\ldots,b^{\prime}_{i_{n-1}}).$

Moreover, the vectors $u_{l}$ correspond to the vectors $v_{l}$ (resp. $v^{\prime}_{l}$) in the chart for $\rho_{\mathscr{X}}$ (resp. $\rho_{\mathscr{X}^{\prime}}$). We now have $v_{0}=-v_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v_{l}$, so that the vectors we are transporting are written on $\tau_{p_{\infty}}$

$$(-v_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v_{l}\,,v_{1},\ldots,v_{n-1})$$

in the chart for $\rho_{\mathscr{X}}$. These are thus mapped to the tuple $(-v^{\prime}_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v^{\prime}_{l},v^{\prime}_{1},\ldots,v^{\prime}_{n-1})$ by the chart for $\rho_{\mathscr{X}^{\prime}}$. We now transport back across $\tau_{C}$ in the chart for $\rho_{\mathscr{X}^{\prime}}$, to get the tuple of vectors

$$(-v^{\prime}_{0}-\sum_{l=1}^{n-1}b^{\prime}_{i_{l}}v^{\prime}_{l}+\sum_{l=1}^{n-1}b_{i_{l}}v^{\prime}_{l}\,,v^{\prime}_{1},\ldots,v^{\prime}_{n-1})$$

according to the relation $-v^{\prime}_{\infty}=-v^{\prime}_{0}-\sum_{l=1}^{n-1}b^{\prime}_{i_{l}}v^{\prime}_{l}$. We now see that after parallel transport the vectors $(u_{0},\ldots,u_{n-1})$ have changed to

$$(u_{0}+\sum_{l=1}^{n-1}(b_{i_{l}}-b^{\prime}_{i_{l}})u_{l}\,,u_{1},\ldots,u_{n-1}),$$

hence the formula Eq. 3.2.3 for the monodromy matrix. ∎