$i = 1,\dots,n$ [@BPR16 5.6.1]. In particular, $\varphi$ induces a fully faithful tropicalization if $\operatorname{trop}_{\varphi}\!|_{\Sigma(\varphi)}$ is injective and all stretching factors are equal to one.

Let $\varphi \colon X  \to Y$ be a closed embedding and let $\Sigma(\varphi)$ be the associated completed extended skeleton. We say that $\operatorname{trop}_{\varphi}$ is a *smooth tropicalization* if it is fully faithful and further for every finite vertex $x$ of $\Sigma(\varphi)$ the primitive integral vectors along the edges adjacent $\operatorname{trop}_{\varphi}(x)$ span a saturated lattice in $N$ of rank $\deg(x) - 1$.

Usually the conditions for smoothness for tropical curves do not reference fully faithfulness and instead weights. This is equivalent to our definition in view of [@J Section 5].

Divisors and rational functions on a metric graph {#sec:skel-div}
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If $\Gamma$ is a $\Lambda$-metric graph then a ($\Lambda$-rational) *divisor* on $\Gamma$ is a finite, formal integer-linear combination of $\Lambda$-rational points on $\Gamma$. These divisors form a free Abelian group, which we call $\operatorname{Div}_\Lambda(\Gamma)$.

A *rational function* on $\Gamma$ is a piecewise-linear function $F$ with integer slopes and such that all the points where $F$ is non-linear are $\Lambda$-rational. If these points where $F$ is non-linear are called $x_1,\dots,x_n$, then the *principal divisor* associated to $F$ 