We know that Picard's group of a compact manifold $X$, denoted $\mathrm{Pic} (X)$ is the group of isomorphism classes of holomorphic line bundles, with the tensor product operation.
We also know that there is a natural isomorphism as follows: $$ \mathrm{Pic} (X) \simeq H^1 (X, \mathcal{O}_{X}^{ \ *} ) $$ I would like to know if it is possible to extend the isomorphism above into a bijection that characterizes the notion of moduli problem as follows : $$ F(T) = \{ \ \text{holomorphic line bundles over} \ \ T \ \} / \sim_{ \mathrm{iso} } \ \simeq \mathrm{Hom} (T , H^1 (X , \mathcal{O}_X^{ \ * } )) $$ such that : $ H^1 (X,\mathcal{O}_X^{ \ * } ) $ is the fine or the coarse moduli space representing the moduli functor $ F $ ? If it is not the case, could you tell me why in details please ?
Thanks in advance for your help.