  In this paper we show that to a unital associative algebra object (resp.
co-unital co-associative co-algebra object) of any abelian monoidal category
$\mathcal{C}$ endowed with a symmetric $2$-trace, one can attach a cyclic
(resp. cocyclic) module, and therefore speak of the cyclic (co)homology of the
(co)algebra "with coefficients in $F$". We observe that if $\mathcal{M}$ is a
$\mathcal{C}$-bimodule category equipped with a stable central pair then
$\mathcal{C}$ acquires a symmetric 2-trace. The dual notions of symmetric
$2$-contratraces and stable central contrapairs are derived as well. As an
application we can recover all Hopf cyclic type (co)homology theories, obtain a
conceptual understanding of anti-Yetter-Drinfeld modules, and give a
formula-free definition of cyclic cohomology. The machinery can also be applied
in settings more general than Hopf algebra modules and comodules.
