  Aims. We investigate the s-process during the AGB phase of stellar models
whose cores are enforced to rotate at rates consistent with asteroseismology
observations of their progenitors and successors. Methods. We calculated new
2M$_{\odot}$, Z=0.01 models, rotating at 0, 125, and 250 km/s at the start of
main sequence. An artificial, additional viscosity was added to enhance the
transport of angular momentum in order to reduce the core rotation rates to be
in agreement with asteroseismology observations. We compared rotation rates of
our models with observed rotation rates during the MS up to the end of core He
burning, and the white dwarf phase. Results. We present nucleosynthesis
calculations for these rotating AGB models that were enforced to match the
asteroseismic constraints on rotation rates of MS, RGB, He-burning, and WD
stars. In particular, we calculated one model that matches the upper limit of
observed rotation rates of core He-burning stars and we also included a model
that rotates one order of magnitude faster than the upper limit of the
observations. The s-process production in both of these models is comparable to
that of non-rotating models. Conclusions. Slowing down the core rotation rate
in stars to match the above mentioned asteroseismic constraints reduces the
rotationally induced mixing processes to the point that they have no effect on
the s-process nucleosynthesis. This result is independent of the initial
rotation rate of the stellar evolution model. However, there are uncertainties
remaining in the treatment of rotation in stellar evolution, which need to be
reduced in order to confirm our conclusions, including the physical nature of
our approach to reduce the core rotation rates of our models, and magnetic
processes.
