TITLE: Natural transformation is a mono iff the components are.
QUESTION [8 upvotes]: Could someone please give an honest proof of the fact that a natural transformation in the functor category $[\mathcal{C},\mathbf{Set}]$ is a monomorpism if and only if each of the components are monomorphisms?
I have seen it in many places, but never seen any proof and embarrasingly I can not do it myself.

REPLY [3 votes]: The complete Q&A thread has been linked by Zhen Lin, but I will reproduce the answer here for your convenience.
1- if all components of a natural transformation are mono (resp. epi), then the natural transformation is mono (resp. epi)
2- if a natural transformation $\tau$ between functors $F,G :\mathcal{C} \to \mathcal{D}$ is mono (resp. epi) and $\mathcal{C}$ is small and $\mathcal{D}$ has pulbacks (resp. pushouts), then each component of $\tau$ is is mono (resp epi).
Statement 1 has been proved by Mossa and Revell in their answers.
Statement 2' s proof can be found in section 2.15 of the Handbook of categorical algebra, volume I (by F. Borceux) pages 87--90. In particular, their Corollary 2.15.3 covers the case for small $\mathcal{C}$ 
Since $Set$ has pullbacks and pushouts, you can apply this theorem to $\tau$ in [$\mathcal{C}$,$Set$]