The universal property of graded $K{K}^G$-theory

A universal category-theoretical characterization of groupoid equivariant $K{K}^G$-theory for ${\mathbb{Z}}_2$-graded $C^{\ast }$-algebras is established, by observing the ``$KK$-axiom’’ that for each $[s,\mathcal{E}\oplus B,\mathbb{F}]\in K{K}^G(A,B)$, the `corner-embedding’ $\ast$-homomorphism $\mathbf{j}:B\to \mathsf{c}\mathsf{l}({\mathcal{K}}_B(\mathcal{E}\oplus B)+s(A)+\mathbb{F}\cdot s(A))$ is invertible in $K{K}^G$. This $KK$-axiom and homotopy-invariance characterize graded $K{K}^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^{\ast }$-algebras via stability, homotopy invariance and splitexactness by Higson.