precisely, let $Coalg({\mathcal{C}})$ denote the subcategory of coassociative counital coalgebra objects of $({\mathcal{C}}, \ot)$. Then $$\begin{aligned}
Fun({\mathcal{C}},{\text{Vec}})&=Fun({\mathcal{C}}_{op},{\text{Vec}}_{op})_{op},\\
\mathcal{Z}_{{\mathcal{C}}}Fun({\mathcal{C}},{\text{Vec}})&\simeq\mathcal{Z}_{{\mathcal{C}}_{op}}Fun({\mathcal{C}},{\text{Vec}})_{op}\\&=\mathcal{Z}_{{\mathcal{C}}_{op}}Fun({\mathcal{C}}_{op},{\text{Vec}}_{op}),\\
Coalg({\mathcal{C}})&=Alg({\mathcal{C}}_{op})_{op}.\end{aligned}$$ Furthermore, a cyclic object in ${\text{Vec}}_{op}$ is the same as a cocyclic object in ${\text{Vec}}$ and we have arrived at the following:

\[trace2\] If $C\in Coalg({\mathcal{C}})$ and $F$ a symmetric 2-trace then $C^\bullet= F(C^{\ot \bullet+1})$ is a cocyclic object in ${\text{Vec}}$.

Recall that for an algebra $A$, we had $$\tau_n = F({\mathfrak{a}}_{1,n}^{n,1})\circ\iota_A(A^{\ot n}).$$ However, after unraveling the above identifications, we have for a coalgebra $C$: $$\tau_n = \iota^{-1}_C(C^{\ot n})\circ F({\mathfrak{a}}_{n,1}^{1,n}).$$

### The contravariant functor $F$.

While the covariant theory discussed above is suitable for explaining the cocyclic structure for the case $C^n=\Hom_H(k, M\ot C^{\ot n+1})$. If we want to deal with the case of $C^n=\Hom_H(M\ot A^{\ot n+1}, k)$ and obtain a cocyclic structure on it, then we need a contravariant $F$. This is not hard to do in light of the above.

We say that a contravariant functor $F$ from ${\mathcal{C}}$ to ${\text{Vec}}$ is a symmetric $2$-contratrace if $F$ is a symmetric $2$-trace on ${\mathcal{C}}_{op}$.

By recalling that $Coalg({\mathcal{C}})=Alg({\mathcal{C}}_{op})_{op}$ we immediately obtain the following:

\[trace3\] If $C\in Coalg({\mathcal{C}})$ and $F$ a symmetric 2-contratrace then $C^\bullet= F(C^{\ot \bullet+1})$ is a cyclic object 