  We use the theory of Tambara modules to extend and generalize the
reconstruction theorem for module categories over a rigid monoidal category to
the non-rigid case. We show a biequivalence between the $2$-category of cyclic
module categories over a monoidal category $\mathscr{C}$ and the bicategory of
algebra and bimodule objects in the category of Tambara modules on
$\mathscr{C}$. Using it, we prove that a cyclic module category can be
reconstructed as the category of certain free module objects in the category of
Tambara modules on $\mathscr{C}$, and give a sufficient condition for its
reconstructability as module objects in $\mathscr{C}$. To that end, we extend
the definition of the Cayley functor to the non-closed case, and show that
Tambara modules give a proarrow equipment for $\mathscr{C}$-module categories,
in which $\mathscr{C}$-module functors are characterized as $1$-morphisms
admitting a right adjoint. Finally, we show that the $2$-category of all
$\mathscr{C}$-module categories embeds into the $2$-category of categories
enriched in Tambara modules on $\mathscr{C}$, giving an ''action via
enrichment'' result.
