Q:

Is the collection of all functions between two sets a set?

Can we say "the set of all functions between two sets" as easily as we could say "the set of all real numbers", for example?

A:

Yes. This is allowed because, set theoretically, functions $A \rightarrow B$ are special subsets of $A \times B$. Sets are closed under cartesian products and comprehension allows you to take arbitrary subsets (as long as you're able to specify the membership condition in your logic).

