Let P be the full subcategory of the category of groups (Grp) whose objects are the perfect groups, i.e. those groups for which every element is a product of commutators (but not necessarily itself a commutator). Then it is a well-known fact that P is a coreflective subcategory of Grp.
Does there exist a monomorphism in P that is not an injective function? If so, is there a full characterization of the monomorphisms in P?
In the case of DivAb, the category of divisible abelian groups, one well-known example of a non-injective monomorphism is the quotient map $\mathbb{Q} \twoheadrightarrow \mathbb{Q}/\mathbb{Z}$. In general, the monomorphisms in DivAb are exactly those for which the kernel in Ab, the category of all abelian groups, is reduced (i.e. has no nontrivial divisible subgroups).
Dually, of course, there are also many categories with non-surjective epimorphisms, with one well-known example being the inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ in the category of rings.
A morphism $f : G \to H$ is a monomorphism iff the kernel of $f$ in the usual sense has trivial commutator subgroup, or equivalently is abelian. Examples of such monomorphisms which aren't injective are given by suitable central extensions. In particular the universal central extension of a perfect group is again perfect. An explicit example is given by the extension $\widetilde{A}_5 \to A_5$ where $\widetilde{A}_5$ is the binary icosahedral group.