We recall from 4.15 that a tropical chart $(V,\varphi_{U})$ consists of an open subset $V$ of ${U^{\rm an}}$ for a very affine open subset $U$ of $X$ such that $V={\rm trop}_{U}^{-1}(\Omega)$ for an open subset $\Omega$ of ${\rm Trop}(U)$. Here, $\varphi_{U}:U\rightarrow T_{U}$ is the canonical moment map. It is a closed embedding to the torus $T_{U}={\rm Spec}(K[M_{U}])$. The tropical variety ${\rm Trop}(U)$ is a tropical cycle of $(N_{U})_{\mathbb{R}}$ and ${\rm trop}_{U}:{U^{\rm an}}\rightarrow(N_{U})_{\mathbb{R}}$ is the tropicalization map. The embedding $\varphi_{U}$ is only determined up to translation by an element in $T_{U}(K)$ and hence the tropical constructions are canonical up to integral $\Gamma$-affine isomorphisms.

Suppose that we have another tropical chart $(V^{\prime},\varphi_{U^{\prime}})$. Then $(V\cap V^{\prime},\varphi_{U\cap U^{\prime}})$ is a tropical chart (see Proposition 4.16) and we get a canonical affine homomorphism $\psi_{U,U\cap U^{\prime}}:T_{U\cap U^{\prime}}\rightarrow T_{U}$ of the underlying tori with $\varphi_{U}=\psi_{U,U\cap U^{\prime}}\circ\varphi_{U\cap U^{\prime}}$ on $U\cap U^{\prime}$ (see 4.12). The associated affine map ${\rm Trop}(\psi_{U,U\cap U^{\prime}}):(N_{U\cap U^{\prime}})_{\mathbb{R}}\rightarrow(N_{U})_{\mathbb{R}}$ maps the tropical variety ${\rm Trop}(U\cap U^{\prime})$ onto ${\rm Trop}(U)$ (use Lemma 4.9). Then we define the restriction of the superform $\alpha\in A^{p,q}({\rm trop}_{U}(V))$ to a superform $\alpha|_{V\cap V^{\prime}}$ on ${\rm trop}_{U\cap U^{\prime}}(V\cap V^{\prime})$ by using the pull-back to ${\rm trop}_{U\cap U^{\prime}}(V\cap V^{\prime})$ with respect to ${\rm Trop}(\psi_{U,U\cap U^{\prime}})$. This plays a crucial role in the following definition: