use this to tropicalize $X$ via $$\operatorname{trop}_{{\varphi}} \coloneqq \operatorname{trop}\circ \varphi^{\operatorname{an}} \colon {X^{\rm an}}\to \operatorname{Trop}(Y).$$ The image of ${X^{\rm an}}$ under $\operatorname{trop}_{{\varphi}}$ is denoted $\operatorname{Trop}_{{\varphi}}(X)$.

Fully faithful, totally faithful and smooth {#sec:ff-and-smooth}
-------------------------------------------

Let $\varphi  \colon X \to Y$ be a map from $X$ to a toric variety $Y$, that is generically finite and whose image meets the dense torus $T$ of $Y$. Let $U \coloneqq \varphi^{-1}(T).$ Let $N$ be the cocharacter lattice of $T$ and $N_{\mathbf{R}}\coloneqq N \otimes_{\operatorname{\mathbf{Z}}} {\mathbf{R}}$. The map $\operatorname{trop}_{\varphi}$ is called *totally faithful* (see [@CFPU]) if it induces an isometry from the associated open skeleton $\mathring \Sigma(\varphi)$ onto its image (which is exactly $\operatorname{trop}({X^{\rm an}}) \cap N_{\mathbf{R}}$.) It is called *fully faithful* if it is further injective when restricted to $\Sigma(\varphi)$. This is equivalent to the statement that $\operatorname{trop}_{\varphi}$ is injective when restricted to $\varphi^{-1}(Y \setminus T)$.

The map $\operatorname{trop}_{\varphi}\!|_{\Sigma(\varphi)}$ is linear with integral slope on each edge of $\Sigma(\varphi)$. We call this slope the *stretching factor* of $\operatorname{trop}_{\varphi}$ on $e$. Identifying $T$ with $\operatorname{\mathbf{G}}_m^n$, the restriction $\varphi_{U}$ is given by rational functions $f_1,\dots,f_n$ on $X$. Then the stretching factors of $\operatorname{trop}_{\varphi}$ on $e$ is given by the gcd of the slopes of $\log \vert f_i \vert|_e$, 