  The equivalence of the discrete isotropic Heisenberg magnet (IHM) model and
the discrete nonlinear Schr\"odinger equation (NLSE) given by Ablowitz and
Ladik is shown. This is used to derive the equivalence of their discretization
with the one by Izergin and Korepin. Moreover a doubly discrete IHM is
presented that is equivalent to Ablowitz' and Ladiks doubly discrete NLSE.
