 Solution

Map a pure, terminating function only when a value is demanded; use the adapters to obtain the cursor form with caching:

def map[T, U](f: T => U, it: Iter[T]): Iter[U] = new Iter[U] {
  def hasNext: Boolean = it.hasNext
  def next(): U = {
    require(it.hasNext, "empty iterator")
    f(it.next())
  }
}

def map[T, U](f: T => U, cur: Cursor[T]): Cursor[U] =
  asCursor(map(f, asIter(cur)))

Each demanded output is the function applied to the corresponding input, preserving order and lazy access even for an infinite sequence. The cursor adapter caches its mapped current value, so repeated reads need not recompute f. Totality/purity of f is the usual assumption for the mathematical sequence-map postcondition; an exception-throwing or state-changing function would need additional behavioral clauses.

Both functions may be named map because their second parameter types differ, Iter[T] versus Cursor[T]. Method overloading selects the applicable signature; its distinction is not based solely on the return type.