Universality transfer by exact gate synthesis

If every generator of a universal quantum gate set has an exact finite implementation over another set, the latter is also universal. For a target unitary and tolerance $\epsilon >0$, first choose an approximating circuit over the universal set, then replace each generator by its exact implementation. The compiled circuit has the same operator and therefore the same approximation error. Overall global phases can be ignored under the usual physically projective convention. Exact synthesis of an already universal family is sufficient; it does not imply exact synthesis of every possible unitary with a finite alphabet of gates.