I am confused by some computations of cyclic bar constructions of monoids. In
KN, proposition 8.1, they claim that ther is a map from the cyclic bar construction associated with a monoid $M$ to the cyclic object $k\mapsto\operatorname{Fun}(S_k,BM)^\sim$ which induces an equivalence on geometric realizations. (I guess that the tilde denotes the maixmal groupoid)