It is obvious from the definitions that the differential forms form a sheaf on ${X^{\rm an}}$. Using the corresponding constructions for superforms on tropical cycles, it is immediate to define the wedge product and differential operators $d$, $d^{\prime}$, $d^{\prime\prime}$ on differential forms on $V$. By 4.6, we have $A^{p,q}(V)=\{0\}$ if $\max(p,q)>\dim(X)$.

For a morphism $\varphi:X^{\prime}\rightarrow X$ and open subsets $V$ (resp. $V^{\prime}$) of ${X^{\rm an}}$ (resp. $(X^{\prime})^{\rm an}$) with $\varphi(V^{\prime})\subset V$, we get a pull-back $\varphi^{*}:A^{p,q}(V)\rightarrow A^{p,q}(V^{\prime})$ defined in the following way: Suppose that $\alpha\in A^{p,q}(V)$ is given by the covering $(V_{i})_{i\in I}$ and the superforms $\alpha_{i}\in A^{p,q}({\rm trop}_{U_{i}}(V_{i}))$ as above. Then there is a covering $(V_{j}^{\prime})_{j\in J}$ of $V^{\prime}$ by tropical charts $(V_{j}^{\prime},\varphi_{U_{j}^{\prime}})$ which is subordinate to $((\varphi^{\rm an})^{-1}(V_{i}))_{i\in I}$. This means that for every $j\in J$, there is $i(j)\in I$ with $V_{j}^{\prime}\subset V_{i(j)}$ and $\varphi(U_{j}^{\prime})\subset U_{i(j)}$ for the corresponding very affine open subsets. Then $\varphi^{*}(\alpha)$ is the differential form on $V^{\prime}$ given by the covering $(V_{j}^{\prime})_{j\in J}$ and the superforms $\varphi^{*}(\alpha_{i(j)})\in A^{p,q}({\rm Trop}(U_{j}^{\prime}))$. We leave the details to the reader. This construction is functorial as usual.