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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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.