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Let $m:B\times B\to B$ be addition, and let $p_1,p_2$ be the projections. Consider the line bundleIts restriction to $B\times \{y\}$ has Picard group class $[T_y^{\ast }L\otimes L^{-1}]={\varphi }_L(y)$; the factor $L_y^{-1}$ is a constant line and has trivial class. Its restriction to $\{0\}\times B$ is also trivial. Choose a trivialization of the fiber $L_0$ and use it to normalize along the zero sections. The defining universal property of the normalized Poincaré line bundle then identifiesMore explicitly, both sides have the same restrictions to all fibers of the second projection, and both are normalized on $\{0\}\times B$; the Seesaw theorem makes their quotient trivial. Here ${\varphi }_L:B\to \hat{B}$ is the homomorphism associated to a line bundle on an abelian variety, and $\hat{B}$ is the dual abelian variety. Pull back by $(f,g)$ to obtain the required identityThe choice of fiber trivialization is immaterial to this identity in the Picard group.

LetIf $L\in {\mathrm{Pic}}^0(X)$, then its restriction to $X\times \{y\}$ is $T_y^{\ast }L\otimes L^{\vee }={\varphi }_L(y)$, up to a constant one-dimensional factor, and is therefore trivial. Its restriction to $\{e\}\times X$ is also trivial. The Seesaw theorem now implies that $\Lambda (L)$ is trivial on $X\times X$.

Conversely, if $\Lambda (L)$ is trivial, restricting it to $X\times \{y\}$ shows that ${\varphi }_L(y)$ is trivial for every $y\in X(k)$. Thus

Part iii already proves injectivity. Pullback along $i_j$ sends translation-invariant line bundles to translation-invariant line bundles, so it definesClearly $\rho \tau =1$.

For $L\in {\mathrm{Pic}}^0(X_1\times X_2)$, put $L_j=i_j^{\ast }L$ andThen $M$ is trivial on both coordinate axes. Since $L$ and the two correcting factors lie in ${\mathrm{Pic}}^0$, translation by $(x_1,e_2)$ leaves $M$ invariant. Hence every restriction $M{|}_{\{x_1\}\times X_2}$ is trivial. The Seesaw theorem and triviality on $X_1\times \{e_2\}$ imply $M\cong {\mathcal{O}}_X$. Thus $L=\tau (L_1,L_2)$, so

The normalized universal line bundle restricts to the class $\xi$ on $A\times \{\xi \}$ and is trivialized on both zero sections. For a line bundle $L$ on an abelian variety, its homomorphism associated to a line bundle on an abelian variety satisfiesas Picard group classes. A choice of trivialization of $L_0$ normalizes the right side; uniqueness then follows from the Seesaw theorem.