adjacent edges is coprime to $\sum_{e : v \in e} s(e)$ and that $\operatorname{trop}_{\varphi}\!|_{\mathring \Sigma(\varphi)}$ is injective. Then the tropicalization induced by $\varphi$ is totally faithful.

Similarly, if $\operatorname{trop}_{\varphi}\!|_{\Sigma(\varphi)}$ is injective, then the tropicalization induced by $\varphi$ is fully faithful.

We have to check that for each domain of linearity of the functions $\log \vert f_i \vert$, the gcd of their slopes is equal to $1$. The extended skeleton $\Sigma$ associated to $\varphi$ is given by taking $\Gamma$ and at each point $c_e, a_e, p_e, q_e, b_e, d_e$ adding a ray $[c_e, c'_e)$ and so on. Note that here it is crucial that we were able to select the points we obtained in Proposition \[D3 lift\] and Proposition \[D1 lift\] and reuse them in Proposition \[D2 lift\], otherwise we would have to potentially add multiple edges.

On the finite edges we have $\log \vert f_i \vert = F_i$, so this can be checked directly (c.f. Figures \[fig:F1-graph\], \[fig:F3-graph\], \[fig:F2-graphT’\] and \[fig:F2-graphT\]. ).

On an infinite edge, $e$, the slope of $\log \vert f_i \vert$ is the coefficient of $D_i$ at the finite endpoint of $e$. So again this can be checked case by case.

The right choice of parameters {#sec:params}
