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We construct a natural transformation between two versions of $G$-equivariant$K$-homology with coefficients in a $G$-$C^{*}$-category for a countablediscrete group $G$. Its domain is a coarse geometric $K$-homology and itstarget is the usual analytic $K$-homology. Following classical terminology, wecall this transformation the Paschke transformation. We show that under certainfiniteness assumptions on a $G$-space $X$, the Paschke transformation is anequivalence on $X$. As an application, we provide a direct comparison of thehomotopy theoretic Davis-L\"uck assembly map with Kasparov's analytic assemblymap appearing in the Baum-Connes conjecture.

Paschke duality and assembly maps