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On the book "Handbook of Categorical Algebra - Vol I" the author writes:

"Again a careless argument would deduce the existence of a category whose objects are the functors from $\mathcal A$ to $\mathcal B$ and whose morphisms are the natural trasnformation between them. But since $\mathcal A$ and $\mathcal B$ have merely classes of objects, there is in general no way to prove the existence of a set of natural transformations between two functors! But when $\mathcal A$ is small, that problem disappears..."

I noticed that in many other sources I've read, the claim of a category of functors between two categories is simply assumed to exist by postulating that functors are the objects, and natural transformations are the morphisms together with the vertical composition...

Hence, my question is how can we construct category of functors when the underlying categories are not small. I'm assuming this is possible, as people usually talk about categories such as $[\mathbf{Set}, \mathbf{Set}]$, where $\mathbf{Set}$ is only locally small.

Moreover, the definition of natural transformations requires indexing $\alpha$ by $a \in Ob(\mathcal C)$. How can we then claim that a natural transformation exists when the domain category $\mathcal C$ has a non-set class of objects?

Davi Barreira
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    Lots of people just don’t care about size issues in category theory (often they can be avoided by “passing to a higher universe” and other boring set theoretic tricks). Some authors don’t mind if we have a class of arrows rather than a set – FShrike Apr 09 '23 at 18:07
  • Can one index on a class of arrows that is not a set? Wouldn't this be wrong? – Davi Barreira Apr 09 '23 at 18:09
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    I don’t know (and don’t care!). None of the texts about / involving category theory that I’ve read have ever said more than ‘whatever, pass to a higher universe’ – FShrike Apr 09 '23 at 18:11
  • I've never come across such statement... In the Handbook of Categorical Algebra the author seems to care about such things, as stated in the question... Can you give me a reference with a more clear explanation on why this issue does not actually matter? – Davi Barreira Apr 09 '23 at 18:14
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    @FShrike "Passing to a higher universe" is not a "boring set theoretic trick", and some of the other workarounds are rather subtle and complicated! – Zhen Lin Apr 09 '23 at 23:19
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    @ZhenLin Sure. My perspective is quite personal, I don’t mean to insult the whole subject of set theory. It just has always seemed to me that size issues are complete non-issues. All I expect from my set theory is that it can describe all the sensible things that I want it to, and all the category theoretic texts I’ve ever read have essentially supported that idea. I rest easy knowing that experts in foundations have worried about it and found adequate solutions, so that I don’t have to. – FShrike Apr 10 '23 at 10:48
  • Experts in foundations have not found adequate solutions, and category theorists by and large are not experts in foundations. On the other hand, experts in foundations who have worried about it have proposed solutions that category theorists do not find adequate. Perhaps you should (re)read Shulman's paper about set theory. – Zhen Lin Apr 10 '23 at 14:06

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First, note that already non-small categories (i.e. such where objects do not form a set) are not 'first-class' objects of the ambient set theory: Instead, they are given by formulas in the meta-theory, describing the respective classes of objects and morphisms. In particular, statements about categories can only be interpreted as meta-theorems "For any formulas $\Phi,\Psi$ describing the objects and morphisms of a category, ..." (I spelled this out in more detail in an old question of mine, Is category theory constructive?)

Things change when you consider 'small' categories, where you either (a) impose that objects and morphisms are sets without further restriction, or (more common) that (b) the set of objects is $\kappa$-small for some suitable choice of cardinal $\kappa$ (cf. the notion of "Grothendieck Universe"). In the latter setting, you can always form categories, functor categories, etc., as first-class (set-based) objects in your ambient set-theory, but you do have to take care of the indexing cardinal.

With the above said, the answer to your question is simple: Yes, without further smallness constraints, you cannot define the category of functors between two non-small categories.

Hanno
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  • Frankly, I didn't quite understand your first two paragraphs. My knowledge on the subject is not that deep... About your last assertion, can one define a category such as the endofunctors from $\mathbf{Set}$ to $\mathbf{Set}$? Why some authors just claim such categories to exist? For example, Leinester's book. – Davi Barreira Apr 09 '23 at 18:20
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    If, working within ZFC, you define $\textbf{Set}$ as the category of all sets, then you can indeed not define the category of endofunctors on it within ZFC. The reason why you don't always see this elaborated is that there is, by now, a well-understood way to work around this, by 'stratifying' the ambient set-theory via the use of Grothendieck universes, thereby allowing you to talk about different levels of "largeness" without ever leaving the realm of sets. – Hanno Apr 09 '23 at 18:26