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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 $\mathbb U$?

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  • $\begingroup$ Other people know better than I, but I think it's because we need a starting-point, as opposed to no particular starting point. Yes, then, it's my impression that it's not magically clear, in general, that all outcomes are independent of choice of universe, but, obviously, we imagine that our "small" projects are independent... This may be incorrect, but, in any case, I am interested to hear what more expert people can say about this. $\endgroup$ May 22 at 3:57

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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 $\mathbb{U}$.

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