Let $\mathbb{T}_{i}\simeq\mathbb{G}_{m}^{n_{i}}$, $i=1,2$, be split tori over $K$, and $\rho\colon\mathbb{T}_{1}\to\mathbb{T}_{2}$ a group morphism. Let $X_{i}$, $i=1,2$, be toric varieties with torus $\mathbb{T}_{i}$. A morphism $\varphi\colon X_{1}\to X_{2}$ is $\rho$-equivariant if the diagram

is commutative. A morphism $\varphi\colon X_{1}\to X_{2}$ is $\rho$-toric if its restriction to $\mathbb{T}_{1}$ agrees with $\rho$. We say that $\varphi$ is equivariant or toric if it is $\rho$-equivariant or $\rho$-toric, respectively, for some $\rho$.