Aspects of equivariant KK-theory in its generators and relations picture

We consider the universal additive category derived from the category of equivariant separable \(C^*\)-algebras by introducing homotopy invariance, stability and split-exactness. We show that morphisms in that category permit a particular simple form, thus obtaining the universal property of \(KK^G\)-theory for G a locally compact group, or a locally compact groupoid with compact base space, or a countable inverse semigroup as a byproduct.