M\ot C^{\ot n+1})\simeq  \Hom_{H}(1, {^{**}C}\ot M\ot C^{\ot n}).$$

If we have an extra condition on $M$, namely that $$\label{centre2}{^{**}C}\ot M   \simeq  M\ot C ,$$ then we obtain the desired $\tau$, i.e., $$\tau_n:\Hom_{H}(1, M\ot C^{\ot n+1})\simeq \Hom_{H}(1, {^{**}C}\ot M\ot C^{\ot n}) \simeq \Hom_{H}(1, M\ot C^{\ot n}\ot C),$$ where we first use the inverse of followed by the inverse of . This suggests that for a rigid monoidal category ${\mathcal{C}}$ we also need $$YD_1({\mathcal{C}})=\{M \in {\mathcal{C}}: {^{**}C}\ot M  \xrightarrow{\sim}   M\ot C , \, \forall C\in {\mathcal{C}}\},$$ with some compatibility conditions, to serve as coefficients. Note that $YD_1$ is not the same as $AYD$, it is “dual" to it.

Preliminaries {#prelim}
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Here we collect some background material that should facilitate the reading of this paper. The content of Sections \[ydsection\] and \[stabilitysection\] is new. The discussion involving Yetter-Drinfeld modules, anti-Yetter-Drinfeld modules, and their generalizations contained in Section \[ydsection\] is especially important. The conceptual reinterpretation of these objects and their associated complicated formulas was one of the motivations for this paper.

(Co)cyclic modules.
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The main goal of this paper is to introduce a suitable categorical language to unify different notions of cyclic homology under a single theory. Therefore we need to recall 