The fictive alterating to Black having the move, in a chess problem (where traditionally White moves first), is called "set play". I assume that's not because you are now allowed to play with the set (of figures) but because the play is set. In some sense. I got no answer on that on a problem chess board; maybe someone here knows definitely what the phrase means?
Addendum: Also, on a related note, if in the set play all Black moves can be answered to fulfil the stipulation, but White really has to search for a useful move, it's called, as a genre, "White to play" (as said, it's always White to play, so its tautological ;-)). Comments on the genesis of that phrase also are welcome.
Nitpick Note: In a helpmate Black begins. Usually.