What are benefits of fixing a Grothendieck universe?

Question

Let's assume the Tarski's axiom: For all set $u$, there is a Grothendieck universe $\mathbb U$ such that $u\in\mathbb U$. From now on I will drop "Grothendieck" and just write "universe." I think what one usually does when doing category theory is that one fixes a universe $\mathbb U$ and consider only categories whose set of morphisms is an element of $\mathbb U$. These categories are said to be $\mathbb U$-small. But I do not get why fixing a universe is necessary. Isn't it just fine to consider only small categories (categories whose set of morphisms is indeed a set under ZFC + Tarski's axiom)? Any $\mathbb U$-small category is also a small category (under ZFC + Tarski's axiom). So why do we need to fix a specific universe $\mathbb U$?

Answer 1

For example, the category of all small categories (or sets, or groups, etc.) is not small. But the category of $\mathbb{U}$-small categories (or sets, or groups, etc.) is small: in particular, it is $\mathbb{U}'$-small for some universe $\mathbb{U}'$ containing $\mathbb{U}$.