Now let $F:N_{\mathbb{R}}^{\prime}\rightarrow N_{\mathbb{R}}$ be an affine map whose underlying linear map is integral, i.e. induced by a homomorphism $N^{\prime}\rightarrow N$. We will define the push-forward of a weighted integral ${\mathbb{R}}$-affine polyhedral complex $({\mathscr{C}}^{\prime},m)$ of pure dimension $n$ on $N_{\mathbb{R}}^{\prime}$. For details, we refer to [AR10], §7. After a subdivision of ${\mathscr{C}}^{\prime}$, we may assume that

$$F_{*}({\mathscr{C}}^{\prime}):=\{F(\sigma^{\prime})\mid\text{$\sigma^{\prime}$ is a face of $\nu^{\prime}\in{\mathscr{C}}^{\prime}$ with $\dim(F(\nu^{\prime}))=n$}\}$$

is a polyhedral complex in $N_{\mathbb{R}}$. We define the multiplicity of an $n$-dimensional $F(\sigma^{\prime})\in F_{*}({\mathscr{C}}^{\prime})$ by

$$m_{F(\sigma^{\prime})}:=\sum_{\nu^{\prime}\in{\mathscr{C}}^{\prime}_{n},\,\nu^{\prime}\subset F^{-1}(F(\sigma^{\prime}))}[N_{\nu^{\prime}}^{\prime}:N_{F(\sigma^{\prime})}]m_{\nu^{\prime}}.$$

Endowed with these multiplicities, we get a weighted integral ${\mathbb{R}}$-affine polyhedral complex $F_{*}({\mathscr{C}}^{\prime},m)$ of $N_{\mathbb{R}}$. If $({\mathscr{C}}^{\prime},m)$ is a tropical cycle, then $F_{*}({\mathscr{C}}^{\prime},m)$ is also a tropical cycle. It might happen that $F_{*}({\mathscr{C}}^{\prime},m)$ is empty, then we get the tropical zero cycle.