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Please check if the following statement is true:

Statement Let $\varphi : X \rightarrow Y$ be a morphism of the category $\mathbf{Set}$. Let $\sim$ be an equivalence relation on $X$. Then $\varphi$ can be factored (in a unique way) $\varphi = u \circ \pi$ where $\pi$ is the projection for this equivalence and $u : (X / \sim) \rightarrow Y$.

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$\varphi$ needs to be constant on equivalence classes for this to hold. – Tobias Kildetoft Oct 26 '13 at 19:02
@TobiasKildetoft: So I suppose my statement is an erroneous formulation of the UMP for coequalizers in Set. What is the correct formulation? – porton Oct 26 '13 at 19:13
I am not sure how to formulate it in category theory terms. If you simply add the assumption that $\varphi$ is constant on equivalence classes, then the result is correct. – Tobias Kildetoft Oct 26 '13 at 19:15
up vote 1 down vote accepted

Your statement is correct , provided that you add that $\varphi$ respects the equivalence,that is: that it is constant on all members of an equivalence class.

You can read about the UMP of coequalizers in Wikipedia

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