By the proof of Theorem B and Proposition 1.5.2 we have the following diagram:

where the upper arrow is an isomorphism of analytic spaces, and the lower one a homeomorphism. Here $\mathfrak{X}_{D}$ and $\mathfrak{N}_{D}$ are the generic fibers (in the sense of Berkovich) of the formal completions $\widehat{\mathscr{X}_{/D}}$ and $\widehat{\mathscr{N}_{/D}}$ respectively, and $\mathrm{Int}(\Sigma_{1})$ denotes the interior of the polyhedral complex $\Sigma_{1}$ obtained by intersecting the fan $\hat{\Sigma}\subset N_{\mathbb{R}}\times\mathbb{R}$ of the normal bundle of $D$ in $\mathscr{X}$ with $N_{\mathbb{R}}\times\{1\}$. In particular, $\mathrm{Int}(\Sigma_{1})$ is embedded in $\Sigma_{D}\simeq N_{\mathbb{R}}\simeq\mathbb{R}^{n}$, the polyhedral decomposition of $\Star(v_{D})$ is the same of $\Sigma_{D}$, and the vertex $v_{D}$ corresponds to the origin. By Section 1.6, the integral affine structure on $\Star(v_{D})$ is the pullback via $\varphi$ of the integral affine structure on $\Sigma_{D}$, and this concludes the proof. ∎