For a point $p\in X_{\Sigma_{2},0}(K)=\mathbb{T}_{2}(K)$, let $t_{p}\colon X_{2}\to X_{2}$ be the morphism induced by the toric action. Denote by $x_{1,0}\in X_{\Sigma_{1}}(K)$ the distinguished point of the principal open subset of $X_{\Sigma_{1}}$. The correspondence $\varphi\mapsto(t_{\varphi(x_{1,0})}^{-1}\circ\varphi,\varphi(x_{1,0}))$ establishes a bijection between the set of equivariant morphisms $\varphi\colon X_{\Sigma_{1}}\to X_{\Sigma_{2}}$ whose image intersects the principal open subset of $X_{\Sigma_{2}}$ and the set of pairs $(\phi,p)$, where $\phi\colon X_{\Sigma_{1}}\to X_{\Sigma_{2}}$ is a toric morphism and $p\in X_{\Sigma_{2},0}(K)$ is a rational point in the principal open subset. Then the result follows from [Oda88, Theorem 1.13]. ∎