Argue by duality that faithful functors also reflect epimorphisms.

Question

I'm working on Category Theory in Context, and I'm stuck on the second part of this problem.

i.e. Argue by duality that faithful functor also reflect epimorphisms.

My attempt:

So I will assume the first part. The most natural way seems to be to consider the contravariant and faithful functor $F: C^{op} \rightarrow D$. Suppose $F f^{op}$ is monomorphism. Then $f^{op} \in C^{op}$ is monomorphism. But this means that $f \in C$ is epimorphism.

Now, it seems if we prove that $F f$ is epimorphism, we would be done. But the problem is that what even would $F f$ be? Since $F$ is a functor from $C^{op}$ to $D$, $F f$ where $f \in C$ wouldn't even make any sense. So I'm stuck here.

Hints would be appreciated, and please note I'm looking to use duality here.

Thanks for your help!

Answer 1

Hint : consider a faithful functor $F$, and look at the same functor but viewed as a functor on opposite categories : $F^{op}: C^{op}\to D^{op}$. It's still faithful, and if $Ff$ is an epimorphism in $D$, then in $D^{op}$...