right-left stable anti Yetter-Drinfeld module (SAYD) over $H$, they assign a cocyclic module structure to $C^n=\Hom_H(M\ot A^{\ot n+1}, k)$. The comodule part of the anti-Yetter-Drinfeld (AYD) module structure appears in the cyclic map $\tau$. The mysterious AYD structure has not been conceptually well-understood in the literature, although it is known that this structure is obtained by replacing the antipode $S$ by $S^{-1}$ in the definition of a Yetter-Drinfeld (YD) module. On the other hand, the YD modules are well-understood as they form the center of the monoidal category of $H$-modules, ${{_H\mathcal{M}}}$. Not only was the categorical meaning of AYD modules not understood, but also it was not clear why such a mysterious structure is needed to obtain a cocyclic module and therefore cyclic cohomology.

### Contravariant cohomology theory.

To answer this question, one can start from scratch and try to define the cyclic map $\tau$ on $C^n=\Hom_H(M\ot A^{\ot n+1}, k)$ directly. This is the only significant addition to the already apparent cosimplicial structure (in the case that $A$ is an algebra, unital and associative of course). More precisely, we need to *slide the first copy of $A$ past $M$ and then to the back.* To understand the idea better let 