In this paper the authors find a complete set of representatives of normalized $3$-cocycles on a finite abelian group $G$. The main point of the paper is the construction (proposition $2.7$) of a chain map from the normalized bar resolution $(B_{\bullet},\partial_{\bullet})$ to some other resolution $(K_{\bullet},d_{\bullet})$. The authors construct a complete set of representatives of normalized $3$-cocycles of $(K_{\bullet},d_{\bullet})$. The authors conclude that this finishes the proof. Now I'm not sure why this holds.
Clearly, the chain map from $B_{\bullet}\rightarrow K_{\bullet}$ induces a map from the cohomology of $K_{\bullet}$ to the cohomology of $B_{\bullet}$ (It's contravariant since we're applying $\text{Hom}_{\mathbb{Z}G}(-,\mathbb{k}^*)$ to the chain complexes). Moreover, since the normalized bar resolution is a projective resolution, the induced map is injective. Hence we can inject the $3$-cocycles of $K_{\bullet}$ into the $3$-cocycles of $B_{\bullet}$, but why would this resulting set be complete? There is probably something easy that I'm missing. Any suggestions would be much appreciated.