Universal property of the graded E-theory functor e

Develop and prove a universal property characterizing the graded E-theory functor e: * → 𝔼 (from graded C^*-algebras to the stable ∞-category 𝔼 of comodules over e^{C_2}(𝒮)), analogous to the known universal characterization of equivariant E-theory for ungraded algebras.

Background

The authors extend E-theory to graded C^*-algebras by constructing a presentably symmetric monoidal stable ∞-category 𝔼 of comodules and a lax symmetric monoidal functor e: * → 𝔼. For ungraded, equivariant E-theory, a universal property (initial among homotopy invariant, K_G-stable, exact, sum-preserving, and s-finitary functors) is established.

For the graded setting, they note that the functor e enjoys analogous formal properties but the existence of a universal characterization is unknown. Establishing such a property would solidify the foundational status of graded E-theory and unify it with the ungraded framework.