Let $A$ be an abelian variety over a field $k$ with group operation $m\colon A\times A\to A$, and let $A'$ be the dual abelian variety. I know that $A'(k)$ is isomorphic to the subgroup $\operatorname{Pic}^0(A)$ of $\operatorname{Pic}(A)$ composed of the line bundles $\mathscr{L}$ satisfying $m^*\mathscr{L}\simeq \mathscr{L}\boxtimes \mathscr{L}$.

Now, let $S$ be a $k$-scheme. Can we give a similar description to $A'(S)$ using the group operation $m_S\colon A_S\times_S A_S\to A_S$ of $A_S = A\times S$? For example, is it true that $A'(S)$ is isomorphic to the subgroup of $\operatorname{Pic}(A_S)$ composed of the line bundles $\mathscr{L}$ satisfying $m_S^*\mathscr{L}\otimes p_S^*e_S^*\mathscr{L}\simeq \mathscr{L}\boxtimes \mathscr{L}$, where $e_S\colon S\to A_S$ is the identity section and $p_S\colon A_S\times_S A_S\to S$ is the structure map?

(I know that $A'(S)$ is the group of isomorphism classes of invertible sheaves on $A_S$ with rigidification along $e_S$ whose fibers are on $\operatorname{Pic}^0$. But I'm looking for a "global", not fiber-wise, description more or less on the same lines as what I propose above.)