The Baum–Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and L¨uck [8] with the category theoretical construction by Meyer and Nest [22]. This extends the result of Hambleton and Pedersen [12] to arbitrary coefficients. Our approach uses abstract properties rather than explicit constructions and is formally similar to Meyer’s and Nest’s identification of their assembly map with the original construction of the assembly map by Baum, Connes and Higson [2].