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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 $Y_1 \times Y_2 \to X_1 \times X_2$? – Zhen Lin Mar 17 '12 at 14:33

By the very definition of a categorical product, $f_i = \pi_i \circ f$ where $f$ is the product of $f_1, f_2$, so it's always possible to recover the morphisms. If one of them is empty, for example $f_1 : Y \rightarrow X_1 \times X_2$ is empty, then it means that $Y = \varnothing$, and the other function must be the empty function too, and $f_1 \times f_2 = \varnothing$ too.

In general, if $Y$ is the initial object of $\mathcal{C}$ (like $\varnothing$ is the initial object of $\mathcal{Set}$), then in the commutative diagram defining the product, the morphisms $f_i : Y \rightarrow X_i$ are the only morphisms from $Y$ to $X_i$, and their product $f$ is the only morphism from $Y$ to the product of the $X_i$.

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Sorry for a stupid question, but why $\pi_i$ are unique? – porton Mar 17 '12 at 14:27
They are not "unique", but they are part of the product: in category theory, the product of $X_1$ and $X_2$ is a object $X$ together with two morphisms $\pi_i : X \rightarrow X_i$ that make the usual commutative diagrams commute. See here for reference: – Najib Idrissi Mar 17 '12 at 14:33

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