  A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class
$\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r:
0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all
continuous functions on $K$, endowed with the supremum norm, for any proper
compact subset $K$ of the unit circle. We prove that this property is invariant
under composition from the left with any non-constant entire function. As an
application, we show that $\mathcal{U}_A (\mathbb{D})$ is strongly-algebrable.
Furthermore, we prove that Abel universality is invariant under composition
from the right with an automorphism $\Phi$ of $\mathbb{D}$ if and only if
$\Phi$ a rotation. On the other hand, we establish the existence of a subset of
$\mathcal{U}_A (\mathbb{D})$ which is residual in the space of holomorphic
functions on $\mathbb{D}$ and is invariant under composition from the right
with any automorphism of $\mathbb{D}$.
