Why the Hom(A,B) is set of Category of sets?

Question

i understood why Category of sets is large(because the collection of objects is proper class).But why is locally small?

Answer 1

Your question boils down to:

Why is the collection of functions from $X$ to $Y$ a set?

To answer this, we need to look at the axioms for sets - that is, ZFC.

First, remember what a function is - it's a set of ordered pairs with a certain property, and an ordered pair is a set of the form $\{\{a\}, \{a, b\}\}$ (this is the Kuratowski definition; there are other alternative definitions of ordered pairs which work, but if you're using one of these you have to say so - Kuratowski is the default). So we'll want to form the set $X\times Y$ of ordered pairs of elements of $X$ and $Y$.

Once we've formed this set, the rest is easy: we'll use powerset and separation to build the set of subsets of $X\times Y$ satisfying the function property. So all we have to do is build $X\times Y$.

Well, $X\times Y$ is a set of subsets of $X\cup Y$ satisfying the ordered pair property - having two elements, one of which is a one-element subset of $X$, the other of which is a two-element set with one element from $X$ and another element from $Y$, such that the two elements of $X$ are the same. So again powerset and separation will solve the problem once we've built $X\cup Y$.

But building $X\cup Y$ is easy: we use pairing to build $\{X, Y\}$, and then union to get $X\cup Y$.

So ZFC proves that the collection of functions from $X$ to $Y$ is a set whenever $X$ and $Y$ are sets. In fact, much less than ZFC was needed here: we only needed powerset, pairing, union, and two particular instances of separation (separation is actually a whole scheme of axioms, rather than a single axiom). This means the argument goes through in a wide variety of settings - in particular, the axiom of choice isn't used here.

Answer 2

A category is said to be locally small if the hom-classes $\mathrm{hom}(X, Y)$ are sets (rather than proper classes). If $X$ and $Y$ are objects in the category $\mathbf{Set}$, $\mathrm{hom}(X, Y)$ comprises the functions from the set $X$ to the set $Y$ and is a set.