= Solution
The bundle is on $A$, so the printed, undefined target $\operatorname{Pic}(X)$ must be read as the <Picard group> $\operatorname{Pic}(A)$. Write this <Picard group> additively and set $q(h)=[h^*L]$. The zero morphism pulls back $L$ to a constant one-dimensional vector space tensored with $\mathcal O_A$, so $q(0)=0$. The <Theorem of the Cube>, pulled back along $(f,g,h):A\to B^3$, gives
$$
q(f+g+h)-q(f+g)-q(f+h)-q(g+h)+q(f)+q(g)+q(h)=0.
$$
This is the <bilinear cross-effect of a line bundle on an abelian variety>: set $d(f,g)=q(f+g)-q(f)-q(g)$. Substitution into this identity gives
$$
d(f+g,h)=d(f,h)+d(g,h).
$$
The definition gives $d(f,g)=d(g,f)$, so additivity holds in both arguments. Additivity includes negative multiples and the zero morphism. \b[Hence $D_L$ is a symmetric <bilinear map> with values in $\operatorname{Pic}(A)$.] The equalities refer to isomorphism classes of <line bundles>; without chosen rigidifications they are not asserted to be specified canonical isomorphisms of bundles.
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