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Consider a category, whose set of objects is a poset.

Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an obvious way.)

Let $\prod f$ is the smallest object $\varphi$ such that $\pi_0$ is a morphism $f_0 \rightarrow \varphi$ and $\pi_1$ is a morphism $f_1 \rightarrow \varphi$.

Can we infer from this that $\prod f$ is a direct product in our category?

If, no, under which additional condition we can infer that $\prod f$ is a direct product in our category?

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What does smallest object mean here? –  Karl Kronenfeld Aug 30 '13 at 20:17
    
@KarlKronenfeld: My error, objects are partially ordered, not morphisms –  porton Aug 30 '13 at 20:18
    
Then, just take the discrete category on the underlying set of some poset. There are no products besides the trivial ones like $f\times f$. One suggestion is to say that if $A\le B$, then there must exist a morphism $A\to B$. –  Karl Kronenfeld Aug 30 '13 at 20:20
    
@KarlKronenfeld: Yes, in the category I am really considering $A\le B$ implies existence of a morphism $A\rightarrow B$. We may add this condition –  porton Aug 30 '13 at 20:25
    
This is known to work for the category of topological spaces. Could you assist me to prove this for topological spaces in an abstract way (that is not just for topological spaces, but for a wider class of categories)? (I don't know the proof even for the special case of topological spaces, please help.) –  porton Aug 30 '13 at 21:53

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