computability – Characterization of computationally universal functions

Is it correct to state that $$u$$ is a universal function if and only if

begin{align} forall f : mathrm{RE} quad exists g : mathrm{R} quad exists h : mathrm{R} quad f = h circ u circ g end{align}
where RE is the set of recursively enumerable functions and R is the set of recursive functions? (Should R be replaced with something like PR?) If so, does anyone know of a reference that states universality in this generic form?