of a 2-trace on $F(m \cdot -)$, i.e., an isomorphism $$\iota_c: F(m\cdot (-\ot c))\simeq F(m\cdot(c\ot -))$$ via the chain of isomorphisms: $$\begin{aligned}
F(m\cdot (-\ot c))&\rightarrow F((m\cdot -)\cdot c)\\
&\rightarrow F(c\cdot(m\cdot -))\\
&\rightarrow F((c\cdot m)\cdot -)\\
&\rightarrow F((m\cdot c)\cdot -)\\
&\rightarrow F(m\cdot(c\ot -))\end{aligned}$$

For the symmetry condition consult the following commutative diagram with all arrows being the obvious isomorphisms:

$$\xymatrix{
F(m\cdot c)\ar[r]\ar[dr]\ar[dd] & F(m\cdot (1\ot c))\ar[d]\\
& F((m\cdot 1)\cdot c)\ar[d]\\
F(c\cdot m)\ar[r]\ar[dr]\ar[dd] & F(c\cdot(m\cdot 1))\ar[d]\\
& F((c\cdot m)\cdot 1)\ar[d]\\
F(m\cdot c)\ar[r]\ar[dr] & F((m\cdot c)\cdot 1)\ar[d]\\
& F(m\cdot (c\otimes 1))\\
}$$ then the first column composes to $Id$ by stability, and the second column composes to $\iota_c(1)$ by definition, the claim follows.

This shows that a stable central pair gives us a symmetric 2-trace and therefore by Propositions \[trace1\] and \[trace2\] produces cyclic and cocyclic objects from algebras and coalgebras. More precisely, $F(m\cdot A^{\ot \bullet+1})$ and $F(m\cdot C^{\ot \bullet+1})$ are cyclic and cocyclic objects, for $A$ an algebra and $C$ a coalgebra respectively. Let us write out the cyclic map for these cases. Roughly speaking, i.e., ignoring the associativity isomorphisms we have $$\tau: F(m\cdot A^{\ot n+1})\xrightarrow{\iota^F_A(m\cdot A^{\ot n})} F(A\cdot m\cdot A^{\ot n})\xrightarrow{F(\iota^m_A\cdot Id^{\ot n})} F(m\cdot A^{\ot n+1})$$ while $$\tau: F(m\cdot C^{\ot n+1})\xrightarrow{F((\iota^m_C)^{-1}\cdot 