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I am working on a project, and I need to know the proof of this:

Any functor which preserves all colimits preserves epimorphisms.

So could you please tell me how or where I can find the proof for this corollary? Thank you very much.

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This follows from the following fact. $e : X \to Y$ is an epimorphism if and only if pushout

is a pushout. This should be very easy to verify. Pushouts are finite colimits and therefore if a functor preserves colimits it will map pushouts to pushouts and therefore epis to epis.

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  • $\begingroup$ @Dan If this answers your question then you should accept it as an answer (click the tick on the left below the downvote button). $\endgroup$ Dec 12, 2013 at 21:01

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