  This is the second part of a two-paper series that establishes the uniqueness
and regularity of a threshold energy wave map that does not scatter in both
time directions.
  Consider the two-sphere valued equivariant energy critical wave maps equation
on 1+2 dimensional Minkowski space, with equivariance class k > 3. It is known
that every topologically trivial wave map with energy less than twice that of
the unique k-equivariant harmonic map Q scatters in both time directions. We
study maps with precisely the threshold energy, i.e., twice the energy of Q.
  In the first part of the series we gave a refined construction of a threshold
wave map that asymptotically decouples into a superposition of two harmonic
maps (bubbles), one of which is concentrating in scale. In this paper, we show
that this solution is the unique (up to the natural invariances of the
equation) two-bubble wave map. Combined with our earlier work we can now give
an exact description of every threshold wave map.
