left-right AYD modules over a Hopf algebra $H$ by $_H\mathcal{AYD}^H$. Note that there are three additional flavors of AYD modules: left-left, right-left, right-right. All of them are equivalent and thus we focus only on the left-right variety. We will need to generalize.

Let $M$ be a left module and a right comodule over $H$, and let $i\in\mathbb{Z}$. We say that $M\in {_H\mathcal{YD}}^H_i$ if $$\label{genyd1}(h{~\hspace{-3pt}^{^{\left(2\right)}}}m){~\hspace{-3pt}_{\left<0\right>}}\ot (h{~\hspace{-3pt}^{^{\left(2\right)}}}m){~\hspace{-3pt}_{\left<1\right>}} S^{-2i}(h{~\hspace{-3pt}^{^{\left(1\right)}}})=h{~\hspace{-3pt}^{^{\left(1\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<1\right>}}.$$ We say that $M$ is a generalized Yetter-Drinfeld module.

The following lemma provides a characterization of the generalized Yetter-Drinfeld modules akin to the one above for the Yetter-Drinfeld modules.

For a Hopf algebra $H$ the generalized $i$th YD condition is equivalent to $$\label{genyd2}
  \rho(hm)=(hm){~\hspace{-3pt}_{\left<0\right>}}\ot(hm){~\hspace{-3pt}_{\left<1\right>}}=h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(3\right)}}}m{~\hspace{-3pt}_{\left<1\right>}} S^{-1-2i}(h{~\hspace{-3pt}^{^{\left(1\right)}}}).$$

Thus ${_H\mathcal{YD}}^H_0={_H\mathcal{YD}}^H$, while ${_H\mathcal{YD}}^H_{-1}={_H\mathcal{AYD}}^H$, and we will also need ${_H\mathcal{YD}}^H_1$ to serve as coefficients of the covariant theory. Note that if instead of ${{_H\mathcal{M}}}$ we consider ${{_H\mathcal{M}}}_{fd}$ then $\mathcal{YD}_1=\mathcal{AYD}^*=\mathcal{AYD}\, contramodules$.

First, we show that $\eqref{genyd2}\Rightarrow\eqref{genyd1}$: $$\begin{aligned}
   (h{~\hspace{-3pt}^{^{\left(2\right)}}}m){~\hspace{-3pt}_{\left<0\right>}}\ot (h{~\hspace{-3pt}^{^{\left(2\right)}}}m){~\hspace{-3pt}_{\left<1\right>}} S^{-2i}(h{~\hspace{-3pt}^{^{\left(1\right)}}})  &=h{~\hspace{-3pt}^{^{\left(3\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(4\right)}}} m{~\hspace{-3pt}_{\left<1\right>}}S^{-1-2i}(h{~\hspace{-3pt}^{^{\left(2\right)}}})S^{-2i}(h{~\hspace{-3pt}^{^{\left(1\right)}}})\\
      &=h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(3\right)}}}m{~\hspace{-3pt}_{\left<1\right>}}\varepsilon(h{~\hspace{-3pt}^{^{\left(1\right)}}})\\
   &=h{~\hspace{-3pt}^{^{\left(1\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<1\right>}}.
 \end{aligned}$$ Now we show that $\eqref{genyd1}\Rightarrow\eqref{genyd2}$: $$\begin{aligned}
   h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(3\right)}}}m{~\hspace{-3pt}_{\left<1\right>}}S^{-1-2i}(h{~\hspace{-3pt}^{^{\left(1\right)}}})   &=(h{~\hspace{-3pt}^{^{\left(3\right)}}}m){~\hspace{-3pt}_{\left<0\right>}}\ot (h{~\hspace{-3pt}^{^{\left(3\right)}}}m){~\hspace{-3pt}_{\left<1\right>}}S^{-2i}(h{~\hspace{-3pt}^{^{\left(2\right)}}})S^{-1-2i}(h{~\hspace{-3pt}^{^{\left(1\right)}}})\\
      &=(h{~\hspace{-3pt}^{^{\left(2\right)}}}m){~\hspace{-3pt}_{\left<0\right>}}\ot (h{~\hspace{-3pt}^{^{\left(2\right)}}}m){~\hspace{-3pt}_{\left<1\right>}}\varepsilon(h{~\hspace{-3pt}^{^{\left(1\right)}}})\\
   &=\varepsilon(h{~\hspace{-3pt}^{^{\left(1\right)}}})\rho(h{~\hspace{-3pt}^{^{\left(2\right)}}}m)\\
   &=\rho(\varepsilon(h{~\hspace{-3pt}^{^{\left(1\right)}}})h{~\hspace{-3pt}^{^{\left(2\right)}}}m)\\
   &=\rho(hm).
 \end{aligned}$$

If ${\mathcal{C}}$ is a monoidal category, ${\mathcal{M}}$ a ${\mathcal{C}}$-bimodule category, and $F: {\mathcal{M}}\rightarrow {\mathcal{M}}$ a monoidal endofunctor, then we use ${\mathcal{M}}^F$ and ${^F{\mathcal{M}}}$ to 