Full & Faithful Functors

Full and faithful Functors $F:\mathbf{A}\to \mathbf{B}$ are respectively surjections and injections on the hom-set restrictions $hom(A,A^{\prime })\stackrel{F}{\to }hom(FA,F{A}^{\prime })\forall A,A^{\prime }\in \mathbf{A}$. But what does this mean exactly? And why isn’t this the same as an isomorphism?

Firstly, note that for a $\mathbf{B}$-morphism $g:B\to B^{\prime }$, there need not exist a $f$ for which $F(f)=g$. $g$ could simply be not a part of the image of $F$.

Let’s consider a $g$ which is a mapping of $F$ i.e. $\exists f$ s.t. $Ff=g$.

Just restricted to these hom-sets, $F$ is a bijection. However, there are other sets in the category and it’s completely posslbe that $\exists f^{\prime }:A_2\to A_2^{\prime }$ s.t. $F{f}^{\prime }=g$. Thus zooming out of our diagram, we get

So once we leave the world of the red-dotted circle that is the hom-set restriction, $F$ needn’t be injective on the whole category.