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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?

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From “a universal object admits the identity morphism on itself” follows another theorem “every endomorphism on every universal object $A$ is equal to $id_A$”. The latter theorem is used in the theorem which you ask about. The proof in those notes is just too scarce. – beroal Aug 26 '11 at 10:44
up vote 5 down vote accepted

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 $A \to X, A \to Y$.

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Perhaps for a beginner, it is worth noting that the defn of universal object also entails that the only automorphism of it respecting other properties is the identity map. This then implies that the two composites @Qiauchu Y. mentions truly must be the respective identities. – paul garrett Aug 23 '11 at 0:35
@Qiaochu Yuan: IMHO your remark is clearly contradicting category theory. :) To resolve this confusion, we should name categories. Every morphism belongs to some category. In case of the product, we consider a category C of 2-sources into $X, Y$. Your “extra data” are actually a part of the object in C. And in C every terminal object in fact only has the trivial automorphism. And there may be other automorphisms, but in Set, not in C. – beroal Aug 25 '11 at 10:02
@beroal: noted. I'll be more precise about what I meant. – Qiaochu Yuan Aug 25 '11 at 16:43

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.

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