  In this paper, we further explore the conceptual approach to cyclic
cohomology with coefficients. In particular we give a derived version of the
definition with better invariance properties. We show that the new definition
agrees with the old under certain conditions and we prove, for the new
definition, both its Morita invariance and its $2$-Morita invariance, under
suitable interpretations of these terms. More generally, we prove a version of
Shapiro's lemma for cyclic cohomology with coefficients.
