  This paper realises the Khovanov homology of a link in the 3-sphere as a
Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the
second author. The starting point is the previously established formality
theorem for the symplectic arc algebra over a field k of characteristic zero.
Here we prove the symplectic cup and cap bimodules which relate different
symplectic arc algebras are themselves formal over k, and construct a long
exact triangle for symplectic Khovanov cohomology. We then prove the symplectic
and combinatorial arc algebras are isomorphic over the integers in a manner
compatible with the cup bimodules. It follows that Khovanov homology and
symplectic Khovanov cohomology co-incide in characteristic zero.
