  In a recent paper [M. Colangeli \textit{et al.}, J.\ Stat.\ Mech.\ P04021,
(2011)] it was argued that the Fluctuation Relation for the phase space
contraction rate $\Lambda$ could suitably be extended to non-reversible
dissipative systems. We strengthen here those arguments, providing analytical
and numerical evidence based on the properties of a simple irreversible
nonequilibrium baker model. We also consider the problem of response, showing
that the transport coefficients are not affected by the irreversibility of the
microscopic dynamics. In addition, we prove that a form of \textit{detailed
balance}, hence of equilibrium, holds in the space of relevant variables,
despite the irreversibility of the phase space dynamics. This corroborates the
idea that the same stochastic description, which arises from a projection onto
a subspace of relevant coordinates, is compatible with quite different
underlying deterministic dynamics. In other words, the details of the
microscopic dynamics are largely irrelevant, for what concerns properties such
as those concerning the Fluctuation Relations, the equilibrium behaviour and
the response to perturbations.
