  For a countable group $G$ we construct a small, idempotent complete,
symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$
whose homotopy category recovers the triangulated equivariant Kasparov category
of separable $G$-$C^*$-algebras, and exhibit its universal property. Likewise,
we consider an associated presentably symmetric monoidal, stable
$\infty$-category $\mathrm{KK}^{G}$ which receives a symmetric monoidal functor
$\mathrm{kk}^{G}$ from possibly non-separable $G$-$C^*$-algebras and discuss
its universal property. In addition to the symmetric monoidal structures, we
construct various change-of-group functors relating these KK-categories for
varying $G$. We use this to define and establish key properties of a (spectrum
valued) equivariant, locally finite $K$-homology theory on proper and locally
compact $G$-topological spaces, allowing for coefficients in arbitrary
$G$-$C^*$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from
$G$-$C^*$-algebras to $G$-$C^*$-categories. These constructions are key in a
companion paper about a form of equivariant Paschke duality and assembly maps.
