I'm looking for the right jargon and notation to represent the following situation:

Let's say you have two functions, one that maps values from $A$ to $B$ and one that maps values from $C$ to $D$.

* How do I denote the set of functions that maps from one set to another set?
* Does a set of functions (in this context) have a standard name?

If I have a function that maps one set of functions to another set of functions, then it seems like there has to be at least two additional functions that map the domain of the first set to the domain of the second, and a second function that maps the range of the first set to the range of the second. These two additional mappings could be injective, surjective or bijective.

* Do these two functions (domain to domain and range to range) have a standard name?
* What $\LaTeX$ arrows should be used to denote the different cases (injective, surjective and bijective)?

Thanks!