I'll try my best to answer your question, but we're on some shaky ground, with regards to foundations of what you call "sets".
However, I've found similar questions are framed better in terms of category theory, which has a much better notion of what you mean by polymorphisms and a "once-inhabited" function space. In the terminology of categories, polymorphic functions are called "natural transformations". There's a bit more going on, but the idea is essentially the same. Similarly, instead of saying "once-inhabited", you'd say that such a natural transformation is "unique up to isomorphism".
Regardless, the way I usually see things like that proved is to let $A$ have only on element. In such a case, there is only one function defined on $A \to A$. So that special case gives you the universal case. A similar technique will probably work for your composition example.
UPDATE:
Here's an example of how I'd prove this using the language of category theory.
Take the category $\textbf{Set}$ of all sets and functions between them. It is well known that the terminal object in $\textbf{Set}$ is the singleton set. In other words, for any set $|A| = 1, A = \{ a \}$, there is at most one function $f : X \to \{ a \}$, for any other set $X$. In other words, $f$ must be the constant function.
Some authors take the empty function ($\emptyset \to \emptyset$) to be a well defined concept (although I personally don't) to make sure that $\textbf{Set}$ is, in fact, a category.
Personally, I find the notion to be a bit silly. If you're coming from a CS background, it's probably closer to the intended semantics to work in $\textbf{Set}_{\bot}$, the category of pointed sets and partially defined functions, since most programs people write are not total functions. And $\bot$ is everywhere you'd want $\emptyset$ to be.
Now consider the function space $\forall A, B, C, (A \to B) \to (B \to C) \to (A \to C)$ where $A, B, C$ are objects in $\textbf{Set}$. To determine the cardinality of this space, as $A, B$ and $C$ vary on $\textbf{Set}$, consider the extreme case where $A, B$ and $C$ are all terminal objects. Then the individual function spaces $f : A \to B$, $g : B \to C$ and $h : A \to C$ all contain, at most, one element. Hence $\forall f, g, h, f \to g \to h$ can, at most, range over one element each: $f, g$, and $h$. Thus, this function space also only has one element.
Side-note
If you find questions like this interesting, and find yourself asking similar questions, I would take some time to read up on categories. They are a lot of fun and can make precise a lot of vague ideas that are not well defined in computer science, making many hard-to-define questions worth asking.