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?
Best Answer
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.