algebraic geometry – Why do we need End(I)=k for neutral Tannakian categories?

I have been reading Milne’s book ‘Basic Theory of Affine Group Schemes’, and in particular the section on Tannaka duality for affine group schemes. The ‘final’ theorem of this section is displayed below:

The conditions in 3.13 are given by

  1. $omega(Xotimes Y) = omega(X) otimes omega(Y)$
  2. There are natural isomorphisms $a_{X,Y,Z}: (Xotimes Y)otimes Z rightarrow Xotimes (Yotimes Z)$ and $tau_{X,Y}: Xotimes Y rightarrow Yotimes X$ whose images under $omega$ are the usual associativity and commutativity isomorphisms of the tensor product of vector spaces
  3. There is an object $Iin textbf{C}$ such that $omega(I) = k$ and isomorphisms $Xotimes I simeq X simeq I otimes X$ whose images are the canonical isomorphisms making $k$ the tensor unit of vector spaces

Because this description of the conditions is a bit lengthily, I also looked into ‘Tannakian Categories’ by Milne and Deligne, and I found the following theorem:

Now clearly the condition of being rigid just makes the difference between affine group scheme and affine monoid scheme. But there also is the condition that $text{End}(I)$ should be the field $k$, and I do not see any counterpart to this in the first exposition. I know that this condition is the same as stating that $I$ should be a simple object, but I do not see any reference to that in the first theorem either.

What am I missing?