  Let H be a coFrobenius Hopf algebra over a field k. Let A be a right
H-comodule algebra over k.
  We recall that the category of right H-comodules admits a certain model
structure whose homotopy category is equivalent to the stable category of right
H-comodules given in Farina's paper. In the first part of this paper, we show
that the category of left A-module objects in the category of right H-comodules
admits a model structure, which becomes a model subcategory of the category of
H*-equivariant A-modules endowed with a model structure given in the author's
previous paper if H is finite dimensional with a certain assumption. Note that
this category is not a Frobenius category in general. We also construct a
functorial cofibrant replacement by proceeding the similar argument as in Qi's
paper. In the latter half of this paper, we see that cyclic H-comodules which
give Hopf-cyclic (co)homology with coefficients in Hopf H-modules are
contructible in the homotopy category of right H-comodules, and we investigate
a Hopf-cyclic (co)homology in slightly modified setting by assuming A a right
H-comodule k-Hopf algebra with H-colinear bijective antipode in stable category
of right H-comodules and give an analogue of the characteristic map. We remark
that, as an expansion of an idea of taking trivial comodule k as the
coefficients, if we take an A-coinvariant part of M assuming M a Hopf A-module
in the category of right H-comodules, we have the degree shift of cyclic
modules.
