> We have more than 50 packages named '*Functions2' which
> implement 'map' between two domains.

I am not suggesting replace those packages that deals with
more than two domains.  I'm talking about
    map : (S->S, %) -> % , where % has a parameter S
There are about 15 map signatures like this.

> Functor is quite loaded name and may lead to confusion.

Yes, the name is a problem.  OpenAxiom uses "Functorial".
But I think Functor is fine, given the context that Haskell is
using this name.

> - What documentatin do you propose for the 'map' function?

>From https://hackage.haskell.org/package/base-4.9.0.0/docs/Data-Functor.html ,
"uniform action over a parameterized type".

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