  In the author's earlier work there appeared a new way to specify any smooth
closed 4-manifold by a surface diagram, which consists of an orientable surface
decorated with simple closed curves. These curves are cyclically indexed, and
each curve has a unique transverse intersection with the next. Each surface
diagram comes from a certain type of map from the 4-manifold to the two-sphere.
The aim of this paper is to give a uniqueness theorem stating that surface
diagrams coming from maps within a fixed homotopy class are unique up to four
moves: stabilization, handleslide, multislide, and shift.
