I have noticed that there is a class of theorems of the form:
Let $A$ be a ring with property $\mathcal P.$ There is a unique ring $A'$ with property $\mathcal Q$ and a homomorphism $i:A\to A'$ such that for any ring $B$ with property $\mathcal Q,$ and a homomorphism $f:A\to B$, there is a unique $g:A'\to B$ such that $g\circ i=f.$
(Here, $\mathcal Q$ is a stronger property than $\mathcal P$.) Some examples of pairs of properties $(\mathcal P,\mathcal Q)$ that work are: $(\mathcal P,\mathcal Q)=$ (integral domain, field), (—, reduced),(field, algebraically closed field), etc.
However, for example, $(\mathcal P,\mathcal Q)=$ (—, integral domain) does not work, as when $A=\mathbb Z/6\mathbb Z,$ $A'=\mathbb Z/2\mathbb Z$ and $A'=\mathbb Z/3\mathbb Z$ both work.
This is obviously not limited to rings, and many examples can be found for groups, sets, etc.
Are there any characterizations of properties $(\mathcal P,\mathcal Q)$ that work?