Full subcategories of $\mathsf{Grp}$ with epis/monos which are not surjective/injective I read that there exist full subcategories of $\mathsf{Grp}$ in which there are epis which are not surjective, and ones in which there are monos that are not injective.
I'm confused because they're full. For instance, a monic $m$ is an arrow which is left cancellable, i.e $$mf=mg\implies f=g$$ Hence, if the subcategory is not full, I can see it's possible for "new monos" to appear, since there are less arrows $f,g$ to check. Injectivity, however, is independent of "context", so an arrow cannot become injective after a change of category. Now  if the subcategory full, how can new monos appear? 
 A: The inclusion $\mathbb{Z} \to \mathbb{Q}$  is an epimorphism in the category of torsion-free abelian groups. This is because if $\mathbb{Q} \to G$ is a homomorphism into a torsion-free abelian group which vanishes on $\mathbb{Z}$, then every element in the image of $\mathbb{Q} \to G$ is annihilated by some positive integer and thus must be zero.
The projection $\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ is a monomorphism in the category of divisible abelian groups. This is because if $f : G \to \mathbb{Q}$ is a homomorphism from a divisible abelian group $G$ which vanishes when composed with  $\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$, its image is a subgroup of $\mathbb{Z}$ and divisible, hence zero.
So here is why there are non-injective monomorphisms and non-surjective epimorphisms: There are fewer test objects which have to satisfy the definining universal property of a monomorphism resp. epimorphism. The observation that $\mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ vanishes shows that $\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ is not a monomorphism in the category of all groups, but $\mathbb{Z}$ is not divisible, so that this does not prevent $\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ to be a monomorphism in the category of divisible abelian groups. Similarly, since $\mathbb{Q}/\mathbb{Z}$ is not torsion-free, it does not prevent $\mathbb{Z} \to \mathbb{Q}$ to be an epimorphism in the category of torsion-free abelian groups.
