Let $A$ be an abelian variety defined over an algebraically closed field, say over $\mathbb{C}$. There is a dual abelian variety $\hat{A}$, along with a Poincare line bundle $L$ on $A\times \hat{A}$. Is there any relation between $\widehat {A\times A}$ and $\hat{A}\times \hat{A}$, for instance are they isogenous. What happens when $A$ is principally polarized, can we say relate the Poincare bundles in this case.