By Proposition 3.1.1 the integral affine structure on $\Star(\tau_{C_{i}})$ identifies $(v_{D_{i-1}},v_{D_{i}},v_{D},v_{D_{i+1}})$ with

$$(v_{0}=(1,0),v_{1}=(0,1),v_{2}=(0,0),v_{\infty}=(-1,b_{i})),$$

while on $\Star(\tau_{C_{i+1}})$ identifies $(v_{D_{i}},v_{D_{i}+1},v_{D},v_{D_{i+2}})$ with

$$(v_{0}=(1,0),v_{1}=(0,1),v_{2}=(0,0),v_{\infty}=(-1,b_{i+1})).$$

It follows that the transition map from the chart $\Star(\tau_{C_{i}})$ to $\Star(\tau_{C_{i+1}})$ of the integral affine structure on $\Star(\tau_{C_{i}})\cap\Star(\tau_{C_{i+1}})$ is given by the matrix $\left(\begin{matrix}b_{i}&1\\ -1&0\end{matrix}\right)$. Thus, the composition of such matrices gives the monodromy around $v_{D}$, along a loop oriented as the path connecting $v_{D_{1}},v_{D_{2}},\ldots,v_{D_{r}},v_{D_{1}}$. ∎