On page 13 of "Categories and Sheaves" by Kashiwara and Schapira, it says that
Note that the category of all sets is not a category since the collection of all sets is not a set. This is one of the reasons why we have to introduce a universe $\mathcal{U}.$
A universe $\mathcal{U}$ is defined to be a set with some properties, one of which is the following:
if $I \in \mathcal{U}$ and $u_i \in \mathcal{U}$ for all $i \in I$, then $\bigcup_{i \in I} u_i \in \mathcal{U}.$
My question: given an arbitrary collection of sets in $\mathcal{U}$ indexed by a set $I$, how do we know that the index set $I$ belongs to $\mathcal{U}$ as well? If the index set always belongs to the universe, then why not change the axiom above to get rid of the assumption that the index set belongs to the universe?
I would appreciate any comments. Thanks.