  The natural problem we approach in the present paper is to show how the
notion of formally smooth (co)algebra inside monoidal categories can substitute
that of (co)separable (co)algebra in the study of splitting bialgebra
homomorphisms. This is performed investigating the relation between formal
smoothness and separability of certain functors and led to other results
related to Hopf algebra theory. Between them we prove that the existence of
$ad$-(co)invariant integrals for a Hopf algebra $H$ is equivalent to the
separability of some forgetful functors. In the finite dimensional case, this
is also equivalent to the separability of the Drinfeld Double $D(H)$ over $H$.
Hopf algebras which are formally smooth as (co)algebras are characterized. We
prove that given a bialgebra surjection $\pi :E\to H$ with nilpotent kernel
such that $H$ is a Hopf algebra which is formally smooth as a $K$-algebra, then
$\pi $ has a section which is a right $H$-colinear algebra homomorphism.
Moreover, if $H$ is also endowed with an $ad$-invariant integral, then this
section can be chosen to be $H$-bicolinear. We also deal with the dual case.
