Let **C**^ = **C***op* to **Set**, for a natural transformation f:X-Y in **C**^, how to prove if f is an epimorphism, then fc is surjective for all objects c in **C**? Anyone can help me with that? Thank you in advance:)



prove natural transformation is epimorphism
It's a good question - possibly homework? - but good homework anyway. 


One thing that makes it a good question is that if you change "epimorphism" to "monomorphism" and "surjective" to "injective" then it becomes much easier: you can solve it directly by testing at representables. (In other words, apply the definition of monomorphism to natural transformations from **C**(-, c) to X, using the Yoneda Lemma.)


But for the question posed, I think you need to apply some fairly serious theory. That's a bit surprising for something that seems so simple, but I don't know any way around it. Andreas's solution gives one very neat method. A slightly different method, which doesn't involve Kan extensions, is to use the theorem that "limits and colimits in a presheaf category are computed pointwise". This roughly speaking means that the forgetful functor from your presheaf category to $\mathbf{Set}^{\mathrm{op} \mathbf{C}}$ creates limits and colimits. And, as in Andreas's solution, you have to apply the (much simpler) fact that a map is an epimorphism if and only if a certain square associated to it is a pushout.


You can find your question on problem sheet 9 of [this course](http://www.maths.gla.ac.uk/~tl/msci/). Brief hints are also given there. The relevant parts of the [associated notes](http://www.maths.gla.ac.uk/~tl/msci/all.pdf) are 4.1.31, 5.1.6 and 5.1.8. 


