The seesaw principle says in general the following.
If $X,T$ are varieties with $X$ complete, and $\mathcal{L}$ a line bundle on $X\times T$, then $$ T_{1} = \{t\in T: \mathcal{L}_{X\times\{t\}} \text{ is trivial on } X\times\{t\}\} $$ is a closed set in $T$. Furthermore, $\mathcal{L}|_{X\times T_{1}}\simeq p_{2}^{*}M$ for some line bundle $M$ on $T_{1}$, where $p_2$ is the natural projection $X\times T_{1}, p_{2}:X\times T_{1}\to T_{1}$.
I was told that one can apply this theorem to prove the following statement, but I haven't figure out how yet. Can anyone give some hints on how to use it? Thank you!
Let $f:A\to B$ be a morphism between abelian varieties, and let $\mathcal P_A$ and $\mathcal P_B$ be the Poincare line bundles of $A$ and $B$ respectively (recall $\mathcal P_A$ is the line bundle on $A\times A^\vee$ such that the degree of its restriction to $A\times\{t\}$ is zero for all $t$, and the restriction to $\{0\}\times A^\vee$ is trivial). Then the dual morphism $f^\vee:B^\vee\to A^\vee$ is the unique map such that $$(id_A,f^\vee)^*\mathcal P_A\simeq (f,id_B)^*\mathcal P_B.$$ This isomorphism is as line bundles defined on $A\times B^\vee$.