If two objects satisfy the same universal property, we know that they are isomorphic in that category. Is the converse true? That is, if two objects are isomorphic in some category, can we construct an appropriate category in which there exists at least one universal property that they both satisfy?
If this is always possible, why can we always construct an appropriate category? If not, what are some examples of isomorphic objects that can't ever be made to satisfy the same universal property?
From universal to isomorphic:
In the category of sets, in which objects are sets and the arrows are functions, then singleton sets are final objects. This means they are isomorphic.
From isomorphic to universal:
Any two sets with the same number of elements are isomorphic in the category of sets, since there is a bijection between them. Can we construct a category in which sets with two elements are final or initial objects?
Definition of Universal Property, for Reference
(Paraphrased from Algebra Chapter 0, by Paolo Aluffi)
An object satisfies a universal property when it is a terminal object of a category. A terminal object is an object that is final, or initial, or both. Let $C$ be a category. An object $I$ of $C$ is initial in $C$ if for every object $A$ of $C$ there exists exactly one morphism from $I$ to $A$ in $C$. An object $F$ of $C$ is final in $C$ if for every object $A$ of $C$ there exists exactly one morphism from $A$ to $F$.