We can define some mathematical objects using universal properties, for example the tensor product, the free group over a set or the Stone–Čech compactification.
I'm wondering about how to develop my intuition so that I can spot a thing I can define using a universal property when I see it.
It seems clear that a necessary condition on the object is that it is unique. For example for two topological spaces and a function $f: X \to Y$ we cannot define continuity of $f$ in terms of universal properties since there are many functions $X \to Y$ that are continuous. But is it sufficient for an object to be unique (up to unique isomorphism) in order for it to be definable using universal properties?
To summarise into one question: what characterises objects that can be defined using universal properties?
Thank you.