Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional left $H$-modules. It becomes a monoidal category by employing the $A$-coring structure of $H$. Note that with this monoidal structure $\operatorname{mod}(H)$ carries a $\ast$-autonomous structure with $\operatorname{Hom}_k(A,k)$ the underlying $k$-vector space of the dualizing object. Let $\operatorname{bimod}(A)$ be the category of $A$-bimodules whose underlying vector space is finite-dimensional.

Denote by $U\colon \operatorname{mod}(H)\rightarrow \operatorname{bimod}(A)$ the forgetful functor. More generally, let $\mathcal{C}$ be an abelian, $k$-linear, essentially small, $\ast$-autonomous category and let $F:\mathcal{C}\rightarrow \operatorname{bimod}(A)$ be a $k$-linear, exact, faithful functor endowed with a strong monoidal structure. Note that for any functor $G$ with target category $\operatorname{bimod}(A)$ the endomomorphism algebra $\operatorname{End}(G)$ is canonically an $A$-bimodule. The left $A$-action is induced by the morphism $$A\rightarrow \operatorname{End}(G); a \mapsto \{\lambda_{G(X)}(a):G(X)\rightarrow G(X)\}_X.$$ Here, $\lambda_{G(X)}\colon A\rightarrow \operatorname{End}(G(X))$ denotes the left action of $A$ on $G(X).$ The right $A$-action on $\operatorname{End}(G)$ is defined analogously.

Question. Is the map $$\operatorname{End}(F)\otimes_A\operatorname{End}(F)\rightarrow \operatorname{End}(F\boxtimes F);\phi\otimes_A \psi\mapsto \{\phi_X\otimes_A \psi_Y\}_{X,Y} $$ an isomorphism of $A$-bimodules? Here, $F\boxtimes F$ denotes the bifunctor $\operatorname{mod}(H) \times \operatorname{mod}(H) \rightarrow \operatorname{bimod}(A)$ given on objects by $(F\boxtimes F)(X,Y)=F(X)\otimes_A F(Y)$. What happens if we consider the strict monoidal forgetful functor $U:\operatorname{mod}(H)\rightarrow \operatorname{bimod}(A)$ for $F$?

Remarks. If one replaces the Hopf algebroid $H$ by a Hopf algebra $H$ over $k$, the category $\operatorname{bimod}(A)$ by the category of finite-dimensional $k$-vector spaces $\operatorname{vect}(k)$ and the functor $F$ by a $k$-linear, exact, faithful functor $G:\operatorname{mod}(H)\rightarrow \operatorname{vect}(k)$ endowed with a strong monoidal structure, one can use coend calculus to show that the $k$-algebras $\operatorname{End}(G)\otimes_A\operatorname{End}(G)$ and $\operatorname{End}(G\boxtimes G)$ are isomorphic; see here. These coend calculations fail for the general Hopf algebroid setting, since $\operatorname{bimod}(A)$ is in general neither rigid (but $\ast$-autonomous) nor symmetric monoidal.

Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional left $H$-modules. It becomes a monoidal category by employing the $A$-coring structure of $H$. Note that with this monoidal structure $\operatorname{mod}(H)$ carries a $\ast$-autonomous structure with $\operatorname{Hom}_k(A,k)$ the underlying $k$-vector space of the dualizing object. Let $\operatorname{bimod}(A)$ be the category of $A$-bimodules whose underlying vector space is finite-dimensional.

Denote by $U\colon \operatorname{mod}(H)\rightarrow \operatorname{bimod}(A)$ the forgetful functor. More generally, let $\mathcal{C}$ be an abelian, $k$-linear, essentially small, $\ast$-autonomous category and let $F:\mathcal{C}\rightarrow \operatorname{bimod}(A)$ be a $k$-linear, exact, faithful functor endowed with a strong monoidal structure. Note that for any functor $G$ with target category $\operatorname{bimod}(A)$ the endomomorphism algebra $\operatorname{End}(G)$ is canonically an $A$-bimodule. The left $A$-action is induced by the morphism $$A\rightarrow \operatorname{End}(G); a \mapsto \{\lambda_{G(X)}(a):G(X)\rightarrow G(X)\}_X.$$ Here, $\lambda_{G(X)}\colon A\rightarrow \operatorname{End}(G(X))$ denotes the left action of $A$ on $G(X).$ The right $A$-action on $\operatorname{End}(G)$ is defined analogously.

Question. Is the map $$\operatorname{End}(F)\otimes_A\operatorname{End}(F)\rightarrow \operatorname{End}(F\boxtimes F);\phi\otimes_A \psi\mapsto \{\phi_X\otimes_A \psi_Y\}_{X,Y} $$ an isomorphism of $A$-bimodules? Here, $F\boxtimes F$ denotes the bifunctor $\mathcal{C}\times \mathcal{C} \rightarrow \operatorname{bimod}(A)$ given on objects by $(F\boxtimes F)(X,Y)=F(X)\otimes_A F(Y)$. What happens if we consider the strict monoidal forgetful functor $U:\operatorname{mod}(H)\rightarrow \operatorname{bimod}(A)$ for $F$?

Remarks. If one replaces the Hopf algebroid $H$ by a Hopf algebra $H$ over $k$, the category $\operatorname{bimod}(A)$ by the category of finite-dimensional $k$-vector spaces $\operatorname{vect}(k)$ and the functor $F$ by a $k$-linear, exact, faithful functor $G:\operatorname{mod}(H)\rightarrow \operatorname{vect}(k)$ endowed with a strong monoidal structure, one can use coend calculus to show that the $k$-algebras $\operatorname{End}(G)\otimes_A\operatorname{End}(G)$ and $\operatorname{End}(G\boxtimes G)$ are isomorphic; see here. These coend calculations fail for the general Hopf algebroid setting, since $\operatorname{bimod}(A)$ is in general neither rigid (but $\ast$-autonomous) nor symmetric monoidal.