$\newcommand{\dmod}{\text{-}\mathrm{mod}}$
Let $A$ be a finite-dimensional $k$-algebra, $A\dmod$ be a category of finite-dimensional A-modules and $\mathrm{U}\_A:A\dmod \to \textbf{Vect}\_k$ be a forgetful functor. We can reconstruct $A$ as $\mathrm{End}(\mathrm{U}\_A)$ by using Tannaka reconstruction thorem.


**Question** : Does the claim hold even if the assumption of "finite-dimensional" is excluded?



Is a smallness condition necessary in the Tannaka reconstruction theorem?
Yes, of course. You still have a natural homomorphism $A\rightarrow END(U\_A)$. Since ${}\_AA$ is a free $A$-module, an endomorphism $x\in END(U\_A)$ is determined by its value $x\_A$ on ${}\_AA$. This proves that the natural homomorphism is an isomorphism:
$$x\_A \in End (A\_{End\_{{}\_AA}})= End (A\_{{A}})=A.$$


