# Why are singletons terminal objects in Set when there can be morphisms between different singletons?

I'm trying to learn category theory and I have a doubt with the Set category. According to Wikipedia https://en.wikipedia.org/wiki/Category_of_sets empty sets are the initial objects, which have morphisms to categories each one containing a set.

It says that singletons are terminal objects but as far as I understand, it's possible to have a morphism from a singleton to another singleton. Say A={e1} and B={e2} are singletons, one could have the morphism <A, f = subset of AxB, B> = <{e1}, {(e1, e2)}, {e2}> which should generate an arrow from {e1} to {e2}, making {e1} non-terminal.

What am I assuming wrong? it it possible that f is instead of a subset of AxB, is a strict subset?

• Terminal doesn't mean there are no morphisms from it. It means that there is only one morphism to it from any given object. "$T$ is terminal if for every object $X$ in $\mathbf C$ there exists a single morphism $X → T$." – Rahul Jan 7 '17 at 17:44
• Yes, I misunderstood what a terminal object is. – Pau Carre Jan 7 '17 at 17:47