Let $\mathbb{T}_{i}$, $N_{i}$, and $\Sigma_{i}$, $i=1,2$, be as above. Then the correspondence $(p,H)\mapsto\varphi_{p,H}$ is a bijection between

1. (1)

   the set of pairs $(p,H)$, where $H\colon N_{1}\to N_{2}$ is a linear map such that for every cone $\sigma_{1}\in\Sigma_{1}$ there exists a cone $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$, and $p$ is a rational point of $X_{\Sigma_{2},0}(K)$,

2. (2)

   the set of equivariant morphisms $\varphi\colon X_{\Sigma_{1}}\to X_{\Sigma_{2}}$ whose image intersects the principal open subset of $X_{\Sigma_{2}}$.