Module version of tensorizing maps and tensor products 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 illustrate our results for the concrete examples of $C^*$-algebras on inverse semigroups.