  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.
