Assume $\mathcal{A}$ is a small cocomplete abelian $\otimes$-category (see [here](https://mathoverflow.net/questions/47079/line-bundles-in-abelian-otimes-categories) for a definition). Is there a cocontinuous, full, faithful, exact $\otimes$-functor $\mathcal{A} \to \text{Mod}(R)$ for some ring $R$?


See [here](https://mathoverflow.net/questions/32173/mitchells-embedding-theorem) for a related question. The reason why I'm asking is that I want to prove that a certain (perfectly natural!) construction works in such a category $\mathcal{A}$ and it seems to me that the axioms of $\mathcal{A}$ are not enough to check that a certain sequence is exact. In module categories there are no problems.



Tensor variant of Mitchell's embedding theorem
Let ${\bf 1}$ denote the unit object of ${\mathcal A}$. If $F: {\mathcal A} \rightarrow Mod(R)$ is a tensor functor, then $F( {\bf 1} ) \simeq R$. If $F$ is fully faithful, then you can recover $R$ as $End({\bf 1})$, and the functor $F$ is given by $A \mapsto Hom( {\bf 1},A)$.
This is rarely a fully faithful embedding. For example, if ${\mathcal A}$ is the category
of complex representations of a finite group, then $F$ annihilates every nontrivial irreducible representation.


