unital associative algebra object. Later we introduce the notion of stable central pair in a monoidal category which is a practical way of obtaining a symmetric $2$-trace.

If ${\mathcal{M}}$ is a ${\mathcal{C}}$-bimodule category then so is $Fun({\mathcal{M}}, {\text{Vec}})$; let ${\mathcal{Z}}_{{\mathcal{C}}}({\mathcal{M}})$ and ${\mathcal{Z}}_{{\mathcal{C}}}Fun({\mathcal{M}}, {\text{Vec}})$ denote their respective centers. If $F\in{\mathcal{Z}}_{{\mathcal{C}}}Fun({\mathcal{M}}, {\text{Vec}})$ and $m\in {\mathcal{Z}}_{{\mathcal{C}}}({\mathcal{M}})$ then the pair $(F, m)$ is called a stable central pair if it satisfies one additional mutual compatibility condition given in Definition \[stablepair\]. We show that any such pair gives us a symmetric $2$-trace.

As an example we see that the monoidal category ${{_H\mathcal{M}}}$ of left modules over a Hopf algebra $H$ can be endowed with a stable central pair and therefore a symmetric $2$-trace. To construct this pair, we consider the ${\mathcal{C}}$ bimodule category ${{^{\#^{-1}}_H\mathcal{M}}}$ where the right ${\mathcal{C}}$-action is given by the monoidal tensor product and the left action is twisted by $S^{-2}$, where $S$ is the antipode of $H$, as we will explain in Section \[ydsection\]. Then if we start with a central element $M\in {\mathcal{Z}}_{{\mathcal{C}}}({{^{\#^{-1}}_H\mathcal{M}}})$, and take $$F:=\Hom_H(1,-)\in {\mathcal{Z}}_{{\mathcal{C}}}Fun({{^{\#^{-1}}_H\mathcal{M}}}, {\text{Vec}}),$$ we observe that $(F, M)$ is a stable central pair (provided that an extra stability condition is satisfied). More interestingly, we prove that 