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I'm trying to understand the categorical definition of a product, which describes them in terms of existence of a unique morphism that makes such-and-such a diagram commute. I don't really feel I've totally understood the motivation for this definition: in particular, why must that morphism be unique? What's the consequence of omitting the requirement for uniqueness in, say, Set?

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    If you omit uniqueness, then any set admitting a surjective map to the usual cartesian product would satisfy the "universal" property.2012-07-10

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