I want to know if there is a problem with this argument:
Let $C$ be a category and $F$ be a faithful functor $C^{op}\rightarrow C$, with left adjoint $F^{op}: C\rightarrow C^{op}$ If $f: A\rightarrow B$ is an epimorphism, this means that $g\circ f=h\circ f\Rightarrow g=h$. Now if I apply $F$ on this, I will get:
$F(g\circ h)=F(h\circ f) \Rightarrow F(g)=F(h)$
this means:
$F(f)\circ F(g)=F(f)\circ F(h)\Rightarrow F(g)=F(h)$
so $F(f)$ is a monomorphism.
Also, let $F(f)$ be a monomorphism, and $F^{op}$ be the left adjoint of $F$ then: $F(f)\circ F(g)=F(f)\circ F(h)\Rightarrow F(g)=F(h)$
$F^{op} F(g\circ f)=F^{op} F(h\circ f)\Rightarrow g=h$
so $f$ is an epimorphism.