  A vector bundle on a projective variety has a natural cohomology if for every
twist its cohomology is concentrated in a single degree. Eisenbud and Schreyer
conjectured there should be vector bundles on $\mathbb{P}^1 \times
\mathbb{P}^1$ with natural cohomology with respect to bundles
$\mathcal{O}(1,0),\mathcal{O}(0,1)$ with prescribed Hilbert polynomial. We
prove this conjecture.
