The full categorical definition of universal morphism was given in Włodzimierz Holsztyński, Universal Mappings and Fixed Points Theorems, Bull Acad Polon Sci, v.XV, No 7, 1967, pp.433-438:
DEFINITION. A morphism $\ u: Y\rightarrow X\ $ in category $\ K\ $ is universal $\ \Leftarrow:\Rightarrow\ $ for any morphism $\ f: Y\rightarrow X\ $ there is an object $\ Z\ $ in $\ K\ $ and a morphism $\ p:Z\rightarrow Y\ $ such that $\ g\circ p=u\circ p.\ $ Object X is stable (or has the fixed morphism property; analogous to the fixed-point property) if the identity $\ i_X:X\rightarrow X\ $ is a universal morphism.
In the topological case one considers the full category of the non-empty topological spaces.
COMMENTS
- There are topological theorems but only one non-obvious general categorical theorem (from late 1960s): Let the finite categorical product $\ \prod_{k=1}^n f_n \ $ of morphisms $\ f:X_{k-1}\rightarrow X_K\ $ be well-defined and universal. Then $\ f_n\circ\ldots\circ f_1 : X_0\rightarrow X_n\ $ is universal too. (This gives examples of universal maps of finite polyhedra which have non-universal product, already in dimension 2). I have a feeling thought that I can generalize the main theorem of the paper mentioned above to address all categories.
- Categories with exactly one object are virtually monoids. Thus in the case of monoids we can talk about universal elements rather than universal morphisms. When $\ 1\ $ is universal then we can say that the monoid itself has the fixed-morphism property. The whole theory here is peculiar since there is only one object. If $\ 1\ $ is not universal then no element is.
- Topic universal maps is a join generalization of the topological dimension theory and of the fixed-point property topic. Some theorems involve--in the same result--the dimension and the fixed point theory (without explicitly mentioning universal maps), and they apply the universal maps in their proof.
(more soon)