$\DeclareMathOperator{\Pic}{Pic}$This question is inspired by my attempt to answer another question. At the end of my answer, there is a missing step, which I don't know how to fill.
Let $X$ be a smooth projective variety over $\mathbb C$. By Hodge theory, the Picard variety $$\Pic^0(X) = H^1(X, \mathcal O_X) / H^1(X, \mathbb Z)$$ is an abelian variety.
Question. Is there a line bundle $\mathcal L$ on $X \times \Pic^0(X)$ such that $$\mathcal L|_{X \times \{L\}} \cong L\tag{1}\label{eq1}$$ for each point $L \in \Pic^0(X)$?
If $X$ is an abelian variety, the Picard variety is the dual abelian variety, $\Pic^0(X) = X^\vee$, and a choice for $\mathcal L$ is the so called Poincaré bundle, see e.g. Wikipedia or [1,2].
According to the nLab, the relative Picard functor is representable, and hence there should be a unique tatological class $[\mathcal L] \in \Pic^0(X \times \Pic^0(X)) / q^*\Pic^0(\Pic^0X)$, where $q: X \times \Pic^0(X) \to \Pic^0(X)$ is the projection map. So any representative $\mathcal L \in \Pic^0(X \times \Pic^0(X))$ of $[\mathcal L]$ should satisfy \eqref{eq1}.
But is there an easier way to construct such a bundle, without showing that $\Pic^0(X)$ represents the relative Picard functor? (Is that even true? nLab doesn't explicitly claim this, and I'm simply going by association here.)
[1] Birkenhake, Lange, Complex abelian varieties
[2] Mumford, Abelian varieties