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If $\mathcal{C}$ is a locally presentable category, then it seems to be well-known that (Strong Epi, Mono) is a factorization system on $\mathcal{C}$. Where can I find a proof of this fact? Actually I only would like to see a proof that every morphism can be factored as an epimorphism followed by a monomorphism.

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This is Proposition 1.61 in [Adámek and Rosický, Locally presentable and accessible categories]. The proof given merely observes that $\mathcal{C}$ is cocomplete, well copowered, and has pullbacks – so it has a (strong epi, mono) factorisation system. Dually, $\mathcal{C}$ has an (epi, strong mono) factorisation system, because it is complete, well powered, and has pushouts. This in turn is Proposition 4.4.3 in [Borceux, Handbook of categorical algebra, Vol. I].

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  • $\begingroup$ Borceux's Proposition 4.4.3 seems to say that complete and well-powered entails (strong epi, mono). Adamek and Rosicky, on the other hand, appear to claim that this instead entails (epi, strong mono), while cocomplete and co-wellpowered entails (strong epi, mono). Only Borceux gives a proof, which seems right to me! This doesn't question the results, of course, any locally presentable category has both kinds of factorisations, but it took me some time to sort things out, so I thought a warning might be helpful. $\endgroup$ – Tom Hirschowitz Feb 28 at 13:46

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