The category of all small categories

Question

Let $CAT$ be the category of all small categories and functors between them. One of the category axioms says that for every two objects $A$ and $B$ of a category, the class of morphisms from $A$ to $B$ must be a set, not a proper class. Assume now that $A$ and $B$ are small categories, that is their classes of objects are sets. How can I deduce that the class of functors from $A$ to $B$ is a set and not a proper class?

Answer 1

First off, the requirement that $\hom(A,B)$ be a small set isn't always assumed; such categories we call "locally small". But $\mathrm{CAT}$ is locally small, so that's a side issue.

It's actually fairly easy to show that $\hom(A,B)$ is small when $A$ and $B$ are small categories: every element can be described by one or more functions between small sets.

And given any two small sets $X$ and $Y$, basic set theory tells us the class of functions from $X$ to $Y$ is a small set.

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For completeness, the encoding I have in mind is to describe a functor $A \to B$ by the two functions describing how it acts on objects of $A$ and how it acts on arrows of $A$.

Other encodings are possible.

Answer 2

A functor $F:A\rightarrow B$ $F_O:Ob(A)\rightarrow Ob(B)$ which is an element of $Set(Ob(A),O(B))$ a map which assign to a morphism $f:X\rightarrow Y$ a morphism $F(f):F(X)\rightarrow F(Y)$, it is a map $F_f: Mor(A)\rightarrow Mor(B)$, remark that $Mor(A)$ is a set since it is $\bigcup_{X,Y\in Ob(A)\times Ob(A)}hom(X,Y)$

Thus the set of functors between $A$ and $B$ can be viewed as a subset of $Ob(B)^{Ob(A)}\times Mor(B)^{Mor(A)}$.

Answer 3

"One of the axioms of category theory says..."?? There are many different axiomatizations! While it is certainly the case that "there must be" a hom-class between any two objects, there is not always an axiom saying so -- Here is one axiomatization that says nothing of the kind! First-order axiom-sets in general do not say anything about their own model theory -- that's a separate semantic issue. The elements of the domain of a model of a first-order theory can be literally anything -- it's the interpretation-functions defined on them (by the model) that matter.