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Barr and Wells state in their text Toposes, Triples and Theories (pdf link)

It seems that no book on category theory is considered complete without some remarks on its set-theoretic foundations. The well-known set theorist Andreas Blass gave a talk (published in Gray [1984]) on the interaction between category theory and set theory in which, among other things, he offered three set-theoretic foundations for category theory. One was the universes of Grothendieck (of which he said that one advantage was that it made measurable cardinals respectable in France) and another was systematic use of the reflection principle, which probably does provide a complete solution to the problem; but his first suggestion, and one that he clearly thought at least reasonable, was: None. This is the point of view we shall adopt.

My question is whether any logical missteps occur by lack of understanding size issues occur when proving statements about categories. For example, the Yoneda Lemma states that for a locally small category $\mathcal C$, $X\in \mathcal C$, and functor $F\colon \mathcal C^{op}\to Set$, $$ Set^{\mathcal C^{op}}(\mathcal C(\square, X),F)\cong FX$$ which is natural in $X$ and $F$.
Aren't we always able to make a category $\mathcal C$ locally $\mathcal U$-small for some Grothendieck universe $\mathcal U$? And if all statements can be made correct using such an expansion of the notion of $Set$, why do we care to mention it?

In other words, should we always be aware of the universe we are working in, or if we should take a Barr and Wells stance and not take such annoyances with much thought? But if it is imperative to be aware, is there an example of an incorrect argument based on such an ignorance of size?

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You can change the universe if you wish, but some problems don't go away. Freyd's theorem that a complete small category must be a poset still holds. More generally, you need to be careful of any word that has an implicit universe parameter. –  Zhen Lin Mar 19 at 8:23

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Without size considerations, you might construct a presheaf with no associated sheaf.

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My question is whether any logical missteps occur by lack of understanding size issues occur when proving statements about categories.

I will give an example. Let $CAT$ denote the "category of all categories". Consider the full subcategory $X\subset CAT$ with objects the categories $C$ such that $Ob(C)\not\in Ob(C)$. Then $Ob(X)\in Ob(X)$ if and only if $Ob(X)\not\in Ob(X)$, reaching a logical problem analogous to Russell's paradox.

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Why does the category of small categories doesn't have the same problem? –  user42912 Apr 12 at 18:27
    
@user42912: because the category of small categories is not small. –  lentic catachresis Apr 12 at 20:47

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