  In this paper, we give a new direct proof of a result by Bobtcheva and
Piergallini that provides finite algebraic presentations of two categories,
denoted $3\mathrm{Cob}$ and $4\mathrm{HB}$, whose morphisms are manifolds of
dimension $3$ and $4$, respectively. More precisely, $3\mathrm{Cob}$ is the
category of connected oriented $3$-dimensional cobordisms between connected
surfaces with connected boundary, while $4\mathrm{HB}$ is the category of
connected oriented $4$-dimensional $2$-handlebodies up to $2$-deformations. For
this purpose, we explicitly construct the inverse of the functor $\Phi:
4\mathrm{Alg} \to 4\mathrm{HB}$, where $4\mathrm{Alg}$ denotes the free
monoidal category generated by a Bobtcheva--Piergallini Hopf algebra. As an
application, we deduce an algebraic presentation of $3\mathrm{Cob}$ and show
that it is equivalent to the one conjectured by Habiro.
