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In category of Set the terminal object is also a separator (or generator). But this is not correct in general category. What condition would guarantee that the terminal object of a category if exists is also a separator? I thought of this one: Any morphism $!:C\rightarrow T $ can be extended uniquely along $u: C \rightarrow A$ for any object $A$. Many Thanks

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Your claim is false even in $\mathbf{Set}$: take $A = \emptyset$. I doubt there is any good characterisation of the terminal object being a separator. –  Zhen Lin May 30 '13 at 22:18
    
Thanks Zhen. But the claim has failed in a very special case (I mean what could possibly be an extension along the `no morphism'). i.e., the unique extension is there as much as the $u:\rightarrow \emptyset$ is there. Can you plz think of any other counterexample or correct me if I am wrong. thanks –  Hooman May 30 '13 at 22:43
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You might be able to get around this glitch by looking at the category of pointed sets (the slice category over a point). –  Sammy Black May 30 '13 at 22:45
    
The category of pointed sets is no example. I agree with Zhen, I doubt that there is any good characterization. And besides, I think that this property is satisfied only in rather boring categories. –  Martin Brandenburg May 30 '13 at 23:03
    
A reformulation could be that the functor $\hom(T,-)$ is faithful (but it does say exactly the same as "T is a separator"). The notion of well-pointed topos can help you in your search. But from what I read, it seems that well-pointed topos is more often sufficient to resemble $\mathbf{Sets}$ than necessary (you can look at Mac Lane and Moerdijk's book Sheaves in Geometry and Logic). –  Pece Jun 2 '13 at 9:12

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