Let $\mathbb{T}_{i}$, $i=1,2$, be split tori over $S$ and $\rho\colon\mathbb{T}_{1}\to\mathbb{T}_{2}$ a morphism of algebraic group schemes. Let ${\mathcal{X}}_{i}$ be toric schemes over $S$ with torus $\mathbb{T}_{i}$ and let $\mu_{i}$ denote the corresponding action. A morphism $\varphi\colon{\mathcal{X}}_{1}\to{\mathcal{X}}_{2}$ is $\rho$-equivariant if the diagram

commutes. A morphism $\varphi\colon{\mathcal{X}}_{1}\to{\mathcal{X}}_{2}$ is $\rho$-toric if its restriction to $\mathbb{T}_{1,\eta}$, the torus over $K$, coincides with that of $\rho$.