Id^{\ot n})} F(C\cdot m\cdot C^{\ot n})\xrightarrow{(\iota^F_C)^{-1}(m\cdot C^{\ot n})} F(m\cdot C^{\ot n+1}),$$ so that in the algebra case $\tau$ moves “last to first" while in the coalgebra case it does the opposite.

### Stable central contrapairs.

Let us now mirror the above discussion for the contravariant case.

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

We immediately obtain:

\[contrapairtocontratrace\] If $(F, m)$ is a stable central contrapair then $F(m \cdot -)$ is a symmetric $2$-contratrace.

Thus as above, $F(m\cdot A^{\ot \bullet+1})$ and $F(m\cdot C^{\ot \bullet+1})$ are cocyclic and cyclic objects, for $A$ an algebra and $C$ a coalgebra respectively. Ignoring the associativity isomorphisms we have $$\tau: F(m\cdot A\ot A^{\ot n})\rightarrow F(A\cdot m\cdot A^{\ot n})\rightarrow F(m\cdot A^{\ot n}\ot A)$$ while $$\tau: F(m\cdot C^{\ot n}\ot C)\rightarrow F(C\cdot m\cdot C^{\ot n})\rightarrow F(m\cdot C\ot C^{\ot n}),$$ so that in the coalgebra case $\tau$ moves “last to first" while in the algebra case it does the opposite.

The monoidal category of left modules over a Hopf algebra
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In this section we apply our 