  We use stripped-down versions of three semi-analytic galaxy formation models
to study the influence of different assumptions about gas cooling and galaxy
mergers. By running the three models on identical sets of merger trees
extracted from high-resolution cosmological N-body simulations, we are able to
perform both statistical analyses and halo-by-halo comparisons. Our study
demonstrates that there is a good statistical agreement between the three
models used here, when operating on the same merger trees, reflecting a general
agreement in the underlying framework for semi-analytic models. We also show,
however, that various assumptions that are commonly adopted to treat gas
cooling and galaxy mergers can lead to significantly different results, at
least in some regimes. In particular, we find that the different models adopted
for gas cooling lead to similar results for mass scales comparable to that of
our own Galaxy. Significant differences, however, arise at larger mass scales.
These are largely (but not entirely) due to different treatments of the `rapid
cooling' regime, and different assumptions about the hot gas distribution. At
this mass regime, the predicted cooling rates can differ up to about one order
of magnitude, with important implications on the relative weight that these
models give to AGN feedback in order to counter-act excessive gas condensation
in relatively massive haloes at low redshift. Different assumptions in the
modelling of galaxy mergers can also result in significant differences in the
timings of mergers, with important consequences for the formation and evolution
of massive galaxies.
