I am trying to understand the concept of duality in category theory, but I am having a problem, well illustrated by the following situation.
Let $H$ be any nontrivial subgroup of the alternating group $G=A_5$. Thus there exists a group monomorphism $H \to G$. By duality, there exists a group epimorphism $G \to H$. This implies $H \cong G/N$ for some normal subgroup $N$ in $G$. This is ridiculous, as $A_5$ is simple. What is wrong here?