  Coin and scattering are the two major formulations for discrete quantum walks
models, each believed to have its own advantages in different applications.
Although they are related in some cases, it was an open question their
equivalence in arbitrary topologies. Here we present a general construction for
the two models for any graph and also for position dependent transition
amplitudes. We then prove constructively their unitary equivalence. Defining
appropriate projector operators, we moreover show how to obtain the
probabilities for one model from the evolution of the other.
