For every $p\in\mathbb H^n$, define the left translation $L_p:\mathbb H^n\to\mathbb H^n$ by $L_p(q)=p\cdot q$. Then $L_p$ is a diffeomorphism, satisfies
for every $q\in\mathbb H^n$. Consequently, under the standard boundary identification $\Phi:\mathbb H^n\to\partial\mathcal U^n$ defined by $\Phi(z,t)=(z,t+i|z|^2)$, each map $\Phi\circ L_p\circ\Phi^{-1}:\partial\mathcal U^n\to\partial\mathcal U^n$ is a CR diffeomorphism.