In the question [Fixed points and universal maps for posets](https://mathoverflow.net/questions/180252/fixed-points-and-universal-maps-for-posets), we find the following definition: if $P, Q$ are partially ordered sets, an order-preserving map $u:P\to Q$ is said to be *universal* if for every order-preserving map $f:P\to Q$ there is $p\in P$ such that $f(p) = u(p)$.


This definition is not poset-specific; it makes sense in a lot of other categories such as $\mathbf{Top}$.


Can we decribe the essence of a universal map in the language of category theory? Is there a universal property that captures universal maps?



"Universal maps" as a universal property
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.



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* 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.


