Let $\varphi^{\prime}:U^{\prime}\rightarrow T^{\prime}$ be a moment map of the non-empty open subset $U^{\prime}$ of $X$ which refines the moment map $\varphi:U\rightarrow T$, i.e. there is an affine homomorphism $\psi:T^{\prime}\rightarrow T$ such that $\varphi=\psi\circ\varphi^{\prime}$ on $U^{\prime}\subset U$. Then we have

$$({\rm Trop}(\psi))_{*}({\rm Trop}(\varphi_{*}^{\prime}(U^{\prime})))={\rm Trop}(\varphi_{*}(U))$$

in the sense of tropical cycles (see 3.9).