We construct a natural transformation between two versions of G-equivariant K-homology with coefficients in a G-C*-category for a countable discrete group G. Its domain is a coarse geometric K-homology and its target is the usual analytic K-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a G-space X, the Paschke transformation is an equivalence on X. As an application, we provide a direct comparison of the homotopy theoretic Davis-Luck assembly map with Kasparov's analytic assembly map appearing in the Baum-Connes conjecture.

Ulrich Bunke was supported by the SFB 1085 (Higher Invariants) funded by the Deutsche Forschungsgemeinschaft (DFG). Alexander Engel acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Priority Programme SPP 2026 "Geometry at Infinity" (EN 1163/5-1, project number 441426261, Macroscopic invariants of manifolds) and through Germany's Excellence Strategy EXC 2044-390685587, Mathematics Munster: Dynamics -- Geometry -- Structure. Markus Land was supported by the research fellowship DFG 424239956, and by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151).