  In this paper, we give a new construction of a Khovanov homotopy type. We
show that this construction gives a space stably homotopy equivalent to the
Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary,
that those two constructions give equivalent spaces. We show that the
construction behaves well with respect to disjoint unions, connected sums and
mirrors, verifying several conjectures from [LS14a]. Finally, combining these
results with computations from [LS14c] and the refined s-invariant from [LS14b]
we obtain new results about the slice genera of certain knots.
