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 (
$\mathscr{C},\otimes$) endowed with a symmetric 2-trace, i.e., an
$F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in
$F$”. Furthermore, we observe that if
$\mathscr{M}$ is a
$\mathscr{C}$-bimodule category and
$(F,M)$ is a stable central pair, i.e.,
$F\in \text{Fun}(\mathscr{M},\text{Vec})$ and
$M\in \mathscr{M}$ satisfy certain conditions, then
$\mathscr{C}$ 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.