  We consider the role of the diffeomorphism constraint in the quantization of
lattice formulations of diffeomorphism invariant theories of connections. It
has been argued that in working with abstract lattices, one automatically takes
care of the diffeomorphism constraint in the quantum theory. We use two systems
in order to show that imposing the diffeomorphism constraint is imperative to
obtain a physically acceptable quantum theory. First, we consider $2+1$ gravity
where an exact lattice formulation is available. Next, general theories of
connections for compact gauge groups are treated, where the quantum theories
are known --for both the continuum and the lattice-- and can be compared.
