as the unit object ${1}$. We apply a similar convention to morphisms. Let $|\vec{n}|=n_1+\cdots+n_k.$ Then for $\vec{n}$ and $\vec{m}$ with $|\vec{n}|=|\vec{m}|$ denote by $${\mathfrak{a}}_{\vec{n}}^{\vec{m}}: A^{\ot\vec{n}}\rightarrow A^{\ot\vec{m}}$$ the unique isomorphism ensured by the monoidal structure. Omitting the brackets enclosing the vector components to reduce clutter, we thus have ${\mathfrak{a}}_{n+1}^{n,1}=Id$, while ${\mathfrak{a}}_{n,1}^{1,n}$ is in general highly non-trivial and will play a central role below. Though ${\mathfrak{a}}_{n,1}^{1,n}$ is invisible for Hopf algebras, it will be needed for quasi-Hopf algebras and similar objects which lack “on the nose" coassociativity.

Symmetric 2-traces
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Let ${\text{Vec}}$ be the category of vector spaces, and $\mathcal{M}$ be a ${\mathcal{C}}$-bimodule category. Then the functor category $Fun(\mathcal{M}, {\text{Vec}})$ is a $\mathcal{C}$-bimodule category with the left and right actions defined by

$$c\cdot F(-):= F(-\cdot c), \quad \text{and} \quad F\cdot c(-):= F(c\cdot -)$$

for all $c\in {\mathcal{C}}$. The center of a ${\mathcal{C}}$-bimodule category $\mathcal{M}$ is denoted by ${\mathcal{Z}}_{{\mathcal{C}}}\mathcal{M}$. Since ${\mathcal{C}}$ is a ${\mathcal{C}}$-bimodule category using its tensor product, we can set $\mathcal{M}={\mathcal{C}}$. To simplify the notation the center of a monoidal category ${\mathcal{C}}$ will be denoted by ${\mathcal{Z}}({\mathcal{C}})$.

Let $(\mathcal{C}, \ot)$ be a monoidal category.

-   A functor $$F\in \mathcal{Z}_{\mathcal{{\mathcal{C}}}}Fun(\mathcal{C}, {\text{Vec}}),$$ is called a 2-trace. In particular we have natural isomorphisms $$\iota_c(-): 