Set union in the category of sets Let $\mathrm{Set}$ be the category of all sets with morphisms being just functions between sets. For two sets $U,V$ there is the categorical notion of their coproduct which is just the disjoint union $U\sqcup V$ or alternatively it can be realised as a colimit. 
Is there a categorical description for $U\cup V$, the usual set union? What I mean is a description which does not refer to the elements of $U$ and $V$.

This is a tentative answer, I'm not sure:
Instead of looking for the union in $Set$, consider $Set_m$, here defined as the subcategory of $Set$ with all sets as elements and only monomorphisms as morphisms. (the singleton is not terminal in $Set_m$, as an unrelated sidenote)
Your $U$ and $V$ objects in $Set$ are also in $Set_m$ so consider them there and take their coproduct in $Set_m$. You get a set $U \cup V$ in $Set_m$ and two inclusion monomorphisms, and you can bring them back into $Set$ by the subcategory inclusion functor.
