in ${\text{Vec}}$.

While $Alg({\mathcal{C}})=Coalg({\mathcal{C}}_{op})_{op}$ implies that:

\[trace4\] If $A\in Alg({\mathcal{C}})$ and $F$ a symmetric 2-contratrace then $C^\bullet= F(A^{\ot \bullet+1})$ is a cocyclic object in ${\text{Vec}}$.

Now for an algebra $A$, we have $$\tau_n = \iota^{-1}_A(A^{\ot n})\circ F({\mathfrak{a}}^{n,1}_{1,n}).$$ Furthermore, for a coalgebra $C$ we get: $$\tau_n = F({\mathfrak{a}}^{1,n}_{n,1})\circ\iota_C(C^{\ot n}).$$

Stable central pairs
--------------------

The concept of a stable central pair introduced in the following definition arises naturally in settings generalizing the Hopf-cyclic theory. The Hopf-cyclic theory itself is implicitly based on it. As the Lemma below demonstrates the reason for its usefulness is that it is a natural way of constructing symmetric $2$-traces, which lead, as we saw above, to cyclic objects.

\[stablepair\] Let $(\mathcal{C}, \ot)$ be a monoidal category, and $\mathcal{M}$ a ${\mathcal{C}}$-bimodule category. Let $F\in Fun (\mathcal{M}, {\text{Vec}})$ and $m\in \mathcal{M}$. The pair $(F, m)$ is called a central pair if

-   $F\in \mathcal{Z}_{{\mathcal{C}}} Fun(\mathcal{M}, {\text{Vec}})$, in particular $\iota^F_c(-): F(-\cdot c)\simeq F(c\cdot -)$.

-   $m\in \mathcal{Z}_{{\mathcal{C}}}(\mathcal{M})$, in particular $\iota^m_c: c\cdot m \simeq m\cdot c$.  

The central pair $(F, m)$ is called a stable central pair if

-   $F(\iota^m_c)\iota^F_c(m)=Id_{F(m\cdot c)}$.

\[pairtotrace\] If $(F, m)$ is a (stable) central pair then $F(m \cdot -)$ is a (symmetric) 2-trace.

Define the structure 