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Suppose we are in a category with morphisms $f,g$ such that $g$ and $fg$ are monomorphisms. Is $f$ also a monomorphism? How do you prove this, assuming it is correct? (which it feels like it should be)

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up vote 4 down vote accepted

No, not necessarily. Since $fg$ mono implies $g$ mono (check!), you're essentially asking if "$fg$ mono implies $f$ mono", which is easily seen to be false, e.g. in $\mathrm{Set}$.

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