It's not a functor; "map" is bind. Here's an example in non-Haskell as I was addressing non-Haskell audience.

  class Maybe {
    value: int;

    static wrap(v: int) { 
        return new Maybe { value = v } 
    }

    // map() applies fn on the unwrapped value. fn returns a Maybe. 
    map(fn: (x: int) => Maybe) {
        return this.value == null ? Maybe.wrap(null) : fn(this.value);
    }
  }

Here Maybe.wrap() is "unit" and Maybe.map() is "bind" in Haskell lingo. The whole thing is a monad. You can chain call with it.

  Maybe.wrap(5)
       .map(x => Maybe.wrap(x + 1))
       .map(x => Maybe.wrap(x * 2))