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I'm reading Goldblatt's: Topoi. I've read about initial and terminal objects and decided to look at the table to think about initial/terminal objects of other categories. There was the category of non-empty sets, does it has an initial object? I think not because any non-empty set can have more than one morphism to another no-empty set, unless they are singletons. And it seems to have a terminal object, which are the singletons. Is this correct?

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Yes, this is exactly right. If $A$ is nonempty and $B$ has more than one point, there is more than one map $A\to B$, so $A$ cannot be initial.

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