The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set→Mon→Grp, where F is the free functor; this functor assigns to every set S the free group on S. The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.
The zero objects in Grp are the trivial groups (consisting of just an identity element).
Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.
Indeed, Grp isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms.
If Grp were an additive category, then this set E of ten elements would be a ring.