 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.