us consider a special case when the monoidal category is rigid, such is the category of *finite* dimensional left modules over a Hopf algebra $H$. Later we will see that the finiteness assumption can be removed. Using the standard adjunction properties of rigidity, for any $V, W\in {{_H\mathcal{M}}}$, we have:

$$\Hom_{H}(V\ot W, 1)\simeq \Hom_{H}(V, 1\ot {W^*})\simeq \Hom_{H}(V, {W^*}\ot 1)\simeq \Hom_H({W^{**}}\ot V, 1).$$

Thus for an $M\in {{_H\mathcal{M}}}$ and an algebra object $A$ (though the algebra structure plays no role at this stage) in ${{_H\mathcal{M}}}$ we obtain

$$\label{adj22} \Hom_{H}(M\ot A^{\ot n+1}, 1)\simeq  \Hom_{H}({A^{**}}\ot M\ot A^{\ot n}, 1).$$

If we have an extra condition on $M$, namely that $$\label{centre22}{A^{**}}\ot M   \simeq  M\ot A ,$$ then we obtain the desired $\tau$, i.e., $$\tau_n:\Hom_{H}(M\ot A^{\ot n+1}, 1)\simeq \Hom_{H}({A^{**}}\ot M\ot A^{\ot n}, 1) \simeq \Hom_{H}(M\ot A^{\ot n}\ot A,1),$$ where we first use , followed by the inverse of . This suggests that for a rigid monoidal category ${\mathcal{C}}$ $$AYD({\mathcal{C}})=\{M \in {\mathcal{C}}: {A^{**}}\ot M  \xrightarrow{\sim}   M\ot A , \, \forall A\in {\mathcal{C}}\},$$ with some compatibility conditions.

Knowing that $YD$ modules form the center of the monoidal category, i.e., $YD=\mathcal{Z}({\mathcal{C}})$, we see that $AYD$ is to $YD$ as ${A^{**}}\ot M \xrightarrow{\sim} M\ot A$ is to $A\ot  M\xrightarrow{\sim} 