# 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

## 1 Answer

Hint: Read the definitions and try to work out what they mean. The first direction, to prove that if it is monic in each of its components is easy. For the other direction, try the following: Try to see if you can find any properties characterising a monomorphism in the category of functors from C to Set. Set has pullbacks, what relates pullbacks and monomorphisms? Construct the pullback of your natural transformation with itself, and check what this implies levelwise.

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Which natural monomorphisms (in general) aren't collections of monomorphism? I'm having trouble thinking of an example. – PyRulez May 18 '15 at 10:44