To any snc model ${\mathcal{X}}$ of $X$ we associated in §2 a hybrid space ${\mathcal{X}}^{\mathrm{hyb}}=X\coprod\Delta({\mathcal{X}})$. Let us briefly recall the topology on ${\mathcal{X}}^{\mathrm{hyb}}$ in the present context. Extend $\pi\colon X\to{\mathbb{D}}^{*}$ to a map

$$\pi\colon{\mathcal{X}}^{\mathrm{hyb}}\to{\mathbb{D}}$$

by declaring $\pi=0$ on $\Delta({\mathcal{X}})$. For $0<r\leq 1$, define ${\mathcal{X}}_{{\mathbb{D}}_{r}}:=\pi^{-1}({\mathbb{D}}_{r})$. The construction in §2 yields, for $0<r\ll 1$, a tropicalization map

$$\log_{\mathcal{X}}\colon{\mathcal{X}}_{{\mathbb{D}}_{r}}\to\Delta({\mathcal{X}})$$

uniquely defined up to an additive error term of size $O((\log|t|)^{-1})$. The topology on ${\mathcal{X}}^{\mathrm{hyb}}$ is the coarsest one such that $\log_{\mathcal{X}}$ is continuous, $\pi$ is continuous, and the inclusion $X\subset{\mathcal{X}}^{\mathrm{hyb}}$ is an open embedding.

Now suppose ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}$ are snc models, with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$. Define the map $\rho^{\mathrm{hyb}}\colon{\mathcal{X}}^{\prime\mathrm{hyb}}\to{\mathcal{X}}^{\mathrm{hyb}}$ to be the identity on $X\subset{\mathcal{X}}^{\prime}$ and equal to the map $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ on $\Delta({\mathcal{X}}^{\prime})$ defined in §4.2.