= Solution
Let $m:B\times B\to B$ be addition, and let $p_1,p_2$ be the projections. Consider the <line bundle>
$$
M=m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}.
$$
Its restriction to $B\times\{y\}$ has <Picard group> class $[T_y^*L\otimes L^{-1}]=\phi_L(y)$; 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 identifies
$$
M\cong(\operatorname{id}_B,\phi_L)^*P_B.
$$
More 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 $\phi_L:B\to\widehat B$ is the <homomorphism associated to a line bundle on an abelian variety>, and $\widehat B$ is the <dual abelian variety>. Pull back by $(f,g)$ to obtain \b[the required identity]
$$
\boxed{D_L(f,g)\cong(f,\phi_L\circ g)^*P_B.}
$$
The choice of fiber trivialization is immaterial to this identity in the <Picard group>.
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