I asked Lumo if he has an answer to this question. He did not like the question too much... ;-) Nevertheless he gave some nice clarifying comments and explained what is wrong with it and how he thinks about the issues mentioned. I think his comments make a very decent answer here anyway (and hope he does not mind that I post them here). So here we go:
Dualities are obviously important and unify several seemingly different descriptions. This is by definition of dualities. In this most general sense, they are analogous to the wave-particle dualism and unification of pictures in quantum mechanics and perhaps other things (unification of electricity and magnetism is substantially different).
The quantum particle is the same thing as the object displaying both wave and particle properties, so the "two" concepts related by the arrow on that line are really the same concept, and the whole relationship claim is vacuous or tautological.
In the same way, the matrix and wave mechanics may be unified but the unification is nothing else than the Dirac formalism for quantum mechanism so the two parts of the relationship are - assuming that the relationship between the pictures is found - a priori equivalent, too. We already have this description for dualities in string theory, sort of, too. One may discuss physics in the description-invariant way. The problem is that we don't have a universal definition of the "Hamiltonian" or "action" but we may still write the general equations with a Hamiltonian or an action that is duality-invariant. This situation differs from the simplest models of quantum mechanics where the Hamiltonian could have been written down "exactly". In string theory, the expressions for the "Hamiltonian" or whatever defines the dynamics depends on the description and it is often incomplete, so the dualities can't be formulated as a sharp mathematical claim at this moment. They're still perfectly true according to all the evidence and tests we may do and assuming it is indeed the case, and it seems to be the case beyond any reasonable doubt, the equivalence is the same equivalence as the equivalence between pictures (Heis/Schr) in quantum mechanics or representations (position/momentum) in quantum mechanics.
Electromagnetism is a bit different because the electromagnetic field contains both the electric vector and the magnetic vector as independent degrees of freedom, so electromagnetism isn't about 2 views on the same 1 thing. It is about 2 things that naturally collaborate and are linked by symmetries and transform into each other under the Lorentz transformations. It's a different relationship than the equivalence in dualities.
Here you can read Lumo's original nice comment.