axioms that specify the type of the Hopf-like object are themselves dictated by exactly this requirement. There are a great many results about recovering the original Hopf-like object from its associated monoidal category provided that some extra structure (a variation on the fiber functor theme) is provided. This explains the important relation between monoidal categories and Hopf algebras. There are also other categorical approaches to cyclic homology [@BS], [@KP]. We observe that our categorical machinery can be applied to the monoidal categories associated to interesting Hopf-like objects such as weak Hopf algebras, Hopf algebroids, quasi-Hopf algebras and Hopfish algebras to obtain homological constructions such as cyclic homology.\
**Acknowledgments**: The authors would like to thank the organizers of the “Noncommutative Geometry Workshop" at the University of Western Ontario, June 2015, where this paper began.

Motivation
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The authors in [@hkrs1] introduced Hopf cyclic cohomology with coefficients for four types of symmetries. In the case of a $H$-module coalgebra $C$, for a right-left stable anti Yetter-Drinfeld module (SAYD) over $H$, they assign a cocyclic module structure to $C^n=\Hom_H(k, M\ot C^{\ot n+1})$. This theory generalizes Connes-Moscovici’s Hopf cyclic cohomology theory [@CM98]. In the case of a $H$-module algebra $A$, and also for a 