In an alternate universe where functional programming was invented by programmers rather than mathematicians, map would probably be called something like convertWith and a functor would probably be called something like a convertible.
If you think of a function as being a way of converting from one kind of thing to another, then a functor is an object which has a number of things which you can convert to some other kind (a number which may be one or zero), using a method of your choice, and map is the name of the method you use to do that.
If you think of a function as being a way of converting from one kind of thing to another, then a functor is an object which has a number of things which you can convert to some other kind (a number which may be one or zero), using a method of your choice, and map is the name of the method you use to do that.