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Suppose that $C$ is a category admitting images. Given an arrow $f:a\to b$ and an epi $e:a'\to a$ is there a common name for a category where (any) of the following properties hold(s)?

1) any image factorization of $f$ gives an image factorization of $f\circ e$

2) any image factorization of $f\circ e$ gives an image factorization of $f$

3) both 1 and 2 hold.

Thanks!

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Taking $f=id$, each condition implies that epis are extremal. For information about extremal epis, see the nlab ncatlab.org/nlab/show/extremal+epimorphism as well as "The Joy of Cats, Abstract and Concrete Categories". I don't know what happens for general $f$. –  Martin Brandenburg Jan 26 '13 at 20:35
    
why not turn this into an answer? –  Ittay Weiss Jan 26 '13 at 20:44
    
Because it is not an answer, right? If every epi is extremal, then I don't see why (1) holds. –  Martin Brandenburg Jan 27 '13 at 0:57
    
sorry @MartinBrandenburg I misread your comment. I need to think about it. –  Ittay Weiss Jan 27 '13 at 2:17

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