Simple Cohomology bundles on multiprojective spaces

We prove stability of the kernel bundle and prove that the cohomology bundle is simple for vector bundles associated to monads on $X=({\mathbb{P}}^{n_1}{)}^2\times \cdots \times ({\mathbb{P}}^{n_s}{)}^2$ for an ample line bundle $\mathcal{L}={\mathcal{O}}_X({\alpha }_1,{\alpha }_1,\cdots ,{\alpha }_s,{\alpha }_s)$.