I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.

**Version 1:**

Let $(C,\otimes)$ be any monoidal category, and I will suppress all associators. By definition, the *abelianization* (or *Hochschild homology*) of $C$ is the (non-monoidal) category $C_{\rm ab}$ (along with a functor $C \to C_{\rm ab}$) formed as follows. The objects of $C_{\rm ab}$ are the same as the objects of $C$. For every pair of objects $X,Y \in C$, you freely include a new morphism $f_{X,Y} : X \otimes Y \overset\sim\to Y\otimes X$ in $C_{\rm ab}$. Let $X,Y,Z$ be any triple of objects in $C$. Then you have new morphisms $f_{X,Y\otimes Z} : X \otimes Y \otimes Z \to Y \otimes Z \otimes X$ and $f_{Y,Z\otimes X} : Z\otimes X \otimes Y$ and $f_{X\otimes Y,Z} : X\otimes Y \otimes Z \to Z\otimes X \otimes Y$; you impose for any such triple the equality $f_{X\otimes Y,Z} = f_{Y,Z\otimes X} \circ f_{X,Y \otimes Z}$. ~~(For example, it follows from this that $f_{X,1} = \operatorname{id}_X = f_{1,X}$ and $f_{X,Y} = f_{Y,X}^{-1}$.)~~ (Edit, following Victor's comment: For example, it follows from this that $f_{1,X} = \operatorname{id}_X$, and $f_{Y,X} \circ f_{X,Y} = f_{X\otimes Y,1}$, which is not required to be the identity.)

One probably should add a few more magic words, and I'm not sure how much follows for free. In general, it's nice to work with categories that are closed under colimits; perhaps we should add those in to $C_{\rm ab}$. And in every monoidal category that I care about, the tensor functor $\otimes$ is cocontinuous in each variable, so probably we should assert that $f_{X,Y}$ is a colimit if $X$ or $Y$ is.

Anyway, modulo these issues, my question is:

Let $H$ be a Hopf algebra and $C = H\text{-mod}$ the monoidal category of its left modules. Is there any reasonable description of $C_{\rm ab}$ in terms of $H$? For example, is $C_{\rm ab}$ the module theory of an algebra $R$ that can be computed directly from $H$?

**Version 2:**

Let $H$ be a Hopf algebra. Following the usual Sweedler notation, denote the comultiplication by $\Delta(a) = \sum a_{(1)}\otimes a_{(2)}$. Then $H \otimes H$ is a left $H$-module by $a \triangleright (h_1\otimes h_2) = \sum a_{(1)}h_1 \otimes a_{(2)}h_2$. Denote this structure by $_H(H\otimes H)$. There is also another left $H$-module structure given by $a \triangleright (h_1\otimes h_2) = \sum a_{(2)}h_1 \otimes a_{(1)}h_2$, which I will denote by $_H^{\rm op}(H\otimes H)$.

What is the universal (in the Morita category of algebras, bimodules, and intertwiners) triple $(R,B,f)$, where $R$ is an algebra and $_R B _H$ is a left $R$- right $H$- bimodule, and $f$ is an isomoprhism of $R$-$(H\otimes H)$-bimodules $$ f : ({_R B_H}) \underset H \otimes ({_H^{\ } (H\otimes H) _{H\otimes H}}) \overset\sim\to ({_R B_H}) \underset H \otimes ({_H^{\rm op} (H\otimes H) _{H\otimes H}}) $$ satisfying a certain equation of the form "$f = f\circ f$" between $R$-$H^{\otimes 3}$-bimodules?

This question is made slightly more complicated by the fact that it's posed in the Morita category of algebras. There might be some abstract nonsense that guarantees that the bimodule $B$ can be assumed to be, say, a rank-one free $R$-module, in which case it's the same as an algebra homomorphism $H \to R$; or perhaps it can be assumed to be a rank-one free $H$-module, in which case it's the same as an algebra homomorphism $R \to H$.