  We study the discretization of a Poincar\'e/Euclidean BF theory. Upon the
addition of a boundary term, this theory is equivalent to the BFCG theory
defined in terms of the Poincar\'e/Euclidean 2-group. At an intermediate step
in the discretization, we note that there are multiple options for how to
proceed. One option brings us back to recovering the discrete variables and
phase space of BF theory. Another option allows us to rediscover the phase
space related to the G-networks given in [2]. Indeed, our main result is that
we are now able to relate the continuum fields with the discrete variables in
[2]. This relation is important to determine how to implement the simplicity
constraints to recover gravity using the BFCG action. In fact, we show that
such relation is not as simple as in the BF discretization: the discretized
variables on the triangles actually depend on several of the continuum fields
instead of solely the continuum B-field.
  We also compare and contrast the discretized BF and BFCG models as pairs of
dual 2-groups. This work highlights (again) how the choice of boundary term
influences the resulting symmetry structure of the \textit{discretized theory}
-- and hence ultimately the choice of quantum states.
