No nonconstant toric morphism from the projective plane to the product of projective lines
= No nonconstant toric morphism from the projective plane to the product of projective lines
Every fan morphism from the fan of $\mathbb P^2$ to the fan of $\mathbb P^1$ is zero: the three source ray images sum to zero, while each adjacent pair must lie in one half-line. Composing a fan morphism to the product fan of $\mathbb P^1\times\mathbb P^1$ with both projections therefore makes it zero.