An object $Q$ in a category $C$ is called cogenerator (coseparator) if for any pair of arrows $f,g \in C(X,Y)$ if $f \neq g$ then there is $h \in C(Y, Q)$ such that $hf \neq hg$. For example, any set with at least two elements is a cogenerator in the category of sets, $\mathbb{Q}/\mathbb{Z}$ is a cogenerator in the category of abelian groups.
It is known that an object $Q$ is a cogenerator iff functor $C(-, Q)$ faithfull (and iff for any object $X$ of $C$ there is set $S$ and a monomorphism $X \to Q^S$).
It looks like there is no cogenerator in the category of groups/monoids/rings. But I wasn't able to prove it. Is it correct? How one can show that?
For rings: If $Q$ were a cogenerator there would have to be a homomorphism $K\to Q$ for every field $K$. But every ring homomorphism from a field is injective, so for every field $K$, $Q$ would have to have a subring isomorphic to $K$. But whatever the cardinality of $Q$ there is a field of larger cardinality.
For groups: If $Q$ were a cogenerator there would have to be a nontrivial homomorphism $G\to Q$ for every simple group $G$. But every nontrivial homomorphism from a simple group is injective, so for every simple group $G$, $Q$ would have to have a subgroup isomorphic to $G$. But whatever the cardinality of $Q$ there is a simple group of larger cardinality.
For monoids: Same as for groups, as a nontrivial monoid homomorphism from a simple group to any monoid is injective.