(2.4) Let $x$ be a point in $X^{\mathrm{an}}$ and let $\mathrm{red}_{\mathscr{X}}(x)$ be its reduction on $\mathscr{X}_{k}$ (see [MN15, 2.2.2]). Let $Z$ be the unique minimal stratum of $\mathscr{X}_{k}$ that contains $\mathrm{red}_{\mathscr{X}}(x)$. By our assumption (2), $Z$ is a log canonical center of $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$. Then $Z\cap\mathscr{X}^{\mathrm{snc}}$ is a non-empty stratum of $\mathscr{X}^{\mathrm{snc}}_{k}$ by the definition of a dlt pair. Thus, it determines a unique face $\tau$ of the dual intersection complex $\Delta(\mathscr{X}^{\mathrm{snc}})$. Let $E_{1},\ldots,E_{r}$ be the prime components of $\mathscr{X}_{k}$ that contain $Z$, and let $N_{1},\ldots,N_{r}$ be their multiplicities in $\mathscr{X}_{k}$. Then $E_{1},\ldots,E_{r}$ correspond precisely to the vertices $v_{1},\ldots,v_{r}$ of $\tau$. We choose a positive integer $m$ such that $mE_{i}$ is Cartier at the point $\mathrm{red}_{\mathscr{X}}(x)$ for every $i$, and we choose a local equation $f_{i}=0$ for $mE_{i}$ at $\mathrm{red}_{\mathscr{X}}(x)$. Then $\rho_{\mathscr{X}}(x)$ is the point of the simplex $\tau$ with barycentric coordinates

$$\alpha=\frac{1}{m}(-N_{1}\ln|f_{1}(x)|,\ldots,-N_{r}\ln|f_{r}(x)|)$$

with respect to the vertices $(v_{1},\ldots,v_{r})$. Under the embedding of $\Delta(\mathscr{X}^{\mathrm{snc}})$ into $X^{\mathrm{an}}$, the point $\rho_{\mathscr{X}}(x)$ corresponds to the monomial point represented by $(\mathscr{X},(E_{1},\ldots,E_{r}),\xi)$ and the tuple

$$\frac{1}{m}(-\ln|f_{1}(x)|,\ldots,-\ln|f_{r}(x)|),$$

in the terminology of [MN15, 2.4.5]. It is obvious that this definition does not depend on the choices of $m$ and the local equations $f_{i}$. It is also straightforward to check that $\rho_{\mathscr{X}}$ is continuous, and that it is a retraction onto $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$.