  Let $\mathcal C$ be a concrete category. We prove that if $\mathcal{C}$
admits a universally free object $\mathsf F$, then there is a projectively
universal morphism $u\colon \mathsf F\to \mathsf F$, i.e., a morphism $u$ such
that for any $B\in \mathcal{C}$ and $\tau\in {\rm Mor}(B)$ there exists an
epimorphism $\pi\in {\rm Mor}(\mathsf F, B)$ such that $\pi \tau = u \pi$. This
builds upon and extends various ideas by Darji and Matheron (Proc. Am. Math.
Soc. 145 (2017)) who proved such a result for the category of separable Banach
spaces with contractive operators as well as certain classes of dynamical
systems on compact metric spaces. Specialising from our abstract setting, we
conclude that the result applies to various categories of Banach
spaces/lattices/algebras, C*-algebras, etc.
