Module tensorizing maps on $ C^*$-algebras

For $C^*$-algebras $\mathfrak{A}, A$ and $B$ where $A$ and $B$ are $\mathfrak{A}$-bimodules with compatible actions, we consider amalgamated $\mathfrak{A}$-module tensor product of $A$ and $B$ and study its relation with the C*-tensor product of $A$ and $B$ for the min and max norms. We introduce and study the notions of module tensorizing maps, module exactness, and module nuclear pairs of $C^*$-algebras in this setting. We give concrete examples of $C^*$-algebras on inverse semigroups.