  Let $f$ be a univalent self-map of the unit disc. We introduce a technique,
that we call {\sl semigroup-fication}, which allows to construct a continuous
semigroup $(\phi_t)$ of holomorphic self-maps of the unit disc whose time one
map $\phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is
of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or
parabolic of zero step) and there is a one-to-one correspondence between the
set of boundary regular fixed points of $f$ with a given multiplier and the
corresponding set for $\phi_1$. Moreover, in case $f$ (and hence $\phi_1$) has
no interior fixed points, the slope of the orbits converging to the
Denjoy-Wolff point is the same. The construction is based on holomorphic
models, localization techniques and Gromov hyperbolicity. As an application of
this construction, we prove that in the non-elliptic case, the orbits of $f$
converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs
domain of $f$ is "almost symmetric" with respect to vertical lines.
