In 1954 Heins showed that if
$$
\varphi_n (z) = \frac{a_n-z}{1-\overline{a_n}z}
$$
is a sequence of conformal automorphisms of the open unit disk $\mathbb{D}$, where $a_n \in \mathbb{D}$ with $a_n \longrightarrow 1$, then there is an infinite Blashcke product $B$ for which the sequence $B\circ \varphi_n$ is dense in the closed unit ball $ \overline{\rm Ball}(H^\infty)$ of $H^\infty$, with the compact-open topology.

More recently, many results along this line were obtained by different authors on $ \overline{\rm Ball}(H^\infty)$. We unify all of these results and give them simplified proofs using a universality criterion that works for a multiplicative topological semigroup.

Motivated by our results, we show that for every nonscalar continuous linear operator $L:H(\mathbb{C})\longrightarrow H(\mathbb{C})$ on the Fr\'echet space $H(\mathbb{C})$ of entire functions that commutes with the differentiation, there is an infinite product of the form
$$
f(z) = \prod_{j =1}^\infty\left(1 - \frac{z}{d_j} \right) \text{ in } H(\mathbb{C}), \text{ where $d_ j \in \mathbb{C}$},
$$
such that the sequence $L^nf $ is dense in $H(\mathbb{C})$. Furthermore, for a sequence of conformal automorphisms $\sigma_n (z) = a_n z+b_n$ of the complex plane $\mathbb{C}$, we offer sufficient conditions, in terms of the coefficients $a_n$ and $b_n$ of $\sigma_n$, for the existence of such a function $f$ for which the sequence $f\circ \sigma_n$ is dense in $H(\mathbb{C})$.