I'm doing a problem to show that any functor $F: \mathcal{A} \to \mathcal{C}$ between categories can be factorized as
$F_L: \mathcal{A} \to \mathcal{B},\,F_R: \mathcal{B} \to \mathcal{C}$, where $F_L$ is a bijection on the objects of $\mathcal{A}$, and $F_R$ is full and faithful.
To some extent, I can sort of see what I'm meant to do, and am trying to use each of $F_L$ and $F_R$ in order to ensure the other has the required properties: for example, if $F$ is not faithful, say $F(f)=F(g)$, then define $F_L(f)=F_L(g)$ so that $F_R F_L(f) = F(f) = F(g) = F_R F_L(g)$ does not contradict faithfulness of $F_R$. However, the only way I could see to make $F_R$ full would be to 'insert' morphisms in $\mathcal{B}$ wherever we have a morphism $c$ in $\mathcal{C}$ which is not mapped onto by $F$, and then map this new morphism under $F_R$ to $c$.
However, doing this raises a number of new problems: first of all, we don't yet know what the objects are in $\mathcal{B}$, though given $F_L$ is bijective we may presumably treat them as the objects of $\mathcal{A}$. Also, we can't just say 'let's add a morphism between any 2 objects in $\mathcal{B}$ wherever $F_R$ needs to map something to another morphism which $F$ is not surjective on', because for one thing we don't know where $F_R$ maps objects to yet, and for another I'm not convinced we can just freely insert morphisms wherever we fancy without problems arising (though I'm new to category theory, so maybe I'm wrong!). Could anyone help? Many thanks - J