  We continue our study on infinitesimal lifting properties of maps between
locally noetherian formal schemes started in math.AG/0604241. In this paper, we
focus on some properties which arise specifically in the formal context. In
this vein, we make a detailed study of the relationship between the
infinitesimal lifting properties of a morphism of formal schemes and those of
the corresponding maps of usual schemes associated to the directed systems that
define the corresponding formal schemes. Among our main results, we obtain the
characterization of completion morphisms as pseudo-closed immersions that are
flat. Also, the local structure of smooth and etale morphisms between locally
noetherian formal schemes is described: the former factors locally as a
completion morphism followed by a smooth adic morphism and the latter as a
completion morphism followed by an etale adic morphism.
