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Can two morphisms (say of the category Set, but also in every category with binary product) be restored knowing a product of these two morphisms?

I'm especially interested in the case if one of the morphisms is empty, e.g. is the morphism from an empty set to an empty set.

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    Your question is unclear. Do you want to understand morphisms $Y \to X_1 \times X_2$, morphisms $X_1 \times X_2 \to Y$, or morphisms $Y_1 \times Y_2 \to X_1 \times X_2$?2012-03-17

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