In the question 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?