We consider the set \({\mathcal {U}}{\mathcal {T}}(A)\) consisting of all scalars \(\lambda \) for which operator \(A-\lambda I\) is universal in the sense of Gian-Carlo Rota. We determine \({\mathcal {U}}{\mathcal {T}}(A)\) in most cases when A is a composition operator on the usual Hardy space \(H^2\) induced by an inner map or the adjoint of that operator. The same problem is considered and solved for select, non-inner, analytic selfmaps of the unit disc and the composition operators induced by them.
