Let $X$, $Y$ be smooth complex projective varieties and $L$ a line bundle on $X\times Y$. Assume that $L|_{X\times \{y\}}\in \mathrm{Pic}^{0}(X)$ for any closed point $y\in Y$ and $L|_{\{x\}\times Y}\in \mathrm{Pic}^{0}(Y)$ for any closed point $x\in X$.
How to show there is a line bundle $M$ on $\mathrm{Alb}(X\times Y)$ such that $L=\mathrm{alb}_{X\times Y}^{*} M?$ (For a smooth proper $\mathbb{C}$-variety $Z$, $\mathrm{alb}_{Z}$ denotes the Albanese map $Z\rightarrow \mathrm{Alb}(Z)=(\mathrm{Pic}_{Z}^0)^{\vee}$.)
By $L|_{X\times\{y\}}\in \mathrm{Pic}^0(X)$, I know that $L$ is the pullback of the poincare line bundle on $X\times \mathrm{Pic}_{X}^0$. Similarly, $L$ is the pullback of the poincare line bundle on $\mathrm{Pic}_{Y}^0\times Y$.