# Natural transformation monomorphism condition

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. –  Zhen Lin Oct 14 '12 at 14:03