I think I understand what you mean. Let's letter a man with the whistle as $M'$, pair of shoes as $S'$ and pair of whistles as $W'$. If we also letter a man without the whistle as $M$ a single shoe as $S$ and a single whistle as $W$, then most people will assume that:
$$S'(S)=2S$$ $$W'(W)=2W$$ $$M'(M,W)=M+W$$
The frame of your question is to challenge the above and claim that any three functions can be used.
Note how the first two functions here depend on one variable, and the last one on two. A function without variables is just a special case of a function with $N$ variables. So we can define $S'$, $W'$ and $M'$ to be constants. Say, $S'$ and $W'$ to be equal zero, and $M'$ to be equal to the desired answer, which can be anything. That gives you the proof, you asked for.
This solution is anticlimactic though. If we go this far, why not also redefine what
= mean or even what
I could guess, that you probably had in mind some way of defining these functions according to some set of rules, that you did not publish ("reasonable" assumptions), so that one can get any result, but since you did not publish these it's impossible to confidently guess what you had in mind as what is "reasonable" often highly subjective. Of course if you are allowed to interpret the puzzle in any way you like you can get any result you like.
In any case, this is hardly what the original author of the puzzle meant.