I have a couple examples of functors that were given to me in class that are faithful, but not full. However, I'd like an actual proof of these facts in case I have to explain myself on an exam.
Example 1: $F:Rings\rightarrow Abelian$ $Groups$ defined by $F(R)=(R,+)$. If $(A,+)$ is an abelian group, define $xy=0$ for $x,y\in A$. The $(A,+)$ is a ring (Could someone please explain to me why this is true?) and $F(A,+,*)=(A,+)$. I'm told this is essentially surjective, faithful, and not full. I'd like an explanation as to why. The more detail the better in case I have to justify myself on an exam.
Example 2: The forgetful functor $F: Groups\rightarrow Sets$ is faithful, but not full. My notes say that each group maps to a unique set and group homomorphism are subsets of the functors. But I don't really understand what that means. Also, my notes claim that $F$ is not full since there exists functions between groups which are not group homomorphisms, but I'd like more explanation on that fact as well.
Please be as detailed as possible with your explanations. I'm very new at Category Theory and I need things broken down for me.
If you have any other examples of faithful, but not full functors please share!