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If I have functors from $C$ to Set for a small category $C$ and a natural transformation between them, how can I show that this natural transformation is monomorphic iff each of its components, indexed by the objects of $C$, is an injection of Set?

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    If a natural transformation is componentwise monic, then it is monic as a natural transformation – this direction is easy. The converse can be proven in several different ways, but you have to use something about the category $\textbf{Set}$. The first one that comes to my mind is to use the fact that $\textbf{Set}$ has equalisers, but this can be done by completely elementary means as well.2012-10-14

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