In Lang’s algebra book, he defines two things.
An object $P$ is called universally attracting if there exists a unique morphism of each object into $P$, and is called universally repelling if for each object, there exists a unique morphism of $P$ into this object. When context makes our meaning clear, we shall objects as above ‘universal’.
Also, he wrote that it is clear that if $P$, $P’$ are two universal objects, then there exists a unique isomorphism between them.
I tried to prove this statement but it wasn’t clear to me.
I’m struggling with the case when $P$ is universally repelling and $P’$ is universally attracting. I think it suffices to show there exists a morphism from $P’$ into $P$.
In the definition of a category, there’s no statement such as [for an ordered pair of objects, there exists at least one a morphism between them]. So in my opinion, it’s possible that there exists a category such that for some pair of objects ($A$, $B$), $Mor(A,B)$ is empty.
How can I prove two universal objects have an isomorphism between them?
Thank you.
As SCappella comments, only for any two universal objects $P,P'$ of the same type (repelling / attracting) there exists a unique isomorphism between them.
If one of $P, P'$ is universally repelling and the other is universally attracting, this is in general not true. In fact, if $P$ is universally repelling and $P'$ is universally attracting, then there exists a unique morphism $\phi : P \to P'$. But there is no reason why $\phi$ should be an isomorphism in any category.
As an example take the category of sets and functions: All one-point sets are universally attracting, but only the empty set is universally repelling.