I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, they ask the following question:


Let $C$ be a $k$-linear rigid abelian tensor category over a field $k$. Let us further assume that $End (\underline{1})=k$ and there exists a fiber functor with values in a field extension $k'\supseteq k$. Does it follow that $C$ is Tannakian?


**What is the current status of this question? A reference to a solution or to a modern discussion would be very useful.**



Saavedra's Definition of Tannakian Category
The answer is yes; in fact, this is mentioned in a footnote to the definition of Tannakian category in Milne's [corrected version of the Deligne-Milne paper,](http://www.jmilne.org/math/xnotes/tc.html) on page 32. This is proved in Deligne's *Catégories tannakiennes*, Theorem 1.12.


