Why are two universal objects isomorphic? Browsing some old online notes, it is claimed that two universal objects are isomorphic because they admit the identity morphism on themselves. But from two identity morphisms, how would you get an isomorphism between the two?
 A: Let $A, B$ be two universal objects. The universal property gives a unique map $A \to B$ compatible with the universal property, and also a unique map $B \to A$ compatible with the universal property. The compositions of these maps in both orders are necessarily $\text{id}_A, \text{id}_B$, again by the universal property. 
Edit: Some remarks are in order. It is not true that a universal object only has trivial automorphisms in the parent category. What is true is that a universal object only has a trivial automorphism compatible with the extra data that comes from being a universal object. For example, if $A$ is the product of objects $X, Y$, this extra data is the pair of projection maps $A \to X, A \to Y$. 
A: For a universal property we can build a category within which the universal object is the initial one. The usual properties of initial objects give the isomorphism you've asked for.
aside:I've never been impressed by the nomenclature universal property, a better one would be a characteristic property, this already carries with it the idea that all objects satisfying the characteristic property are isomorphic because they are characterised by it. 
