Relation between monomorphicity and faithfulness? I'm a beginner in cat theory, so this may seem straight-forward, but I didn't manage to untangle anything by trying to write it calmly...
This question arised from the fact that in Cat, whose objetcts are small cats and arrows functors betweens those cats, the isomorphisms are precisely the isomorphisms of categories (isomorphic functors).
Also in Cat, there is a notion of monomorphicity that applies to functors, since they are precisely the morphisms of Cat.
But we also have a notion of faithfulness of a functor. In this special case of morphisms of Cat, how do these notions relate one to each other ?
The real question is : if a functor between small cats is interpreted as a morphism of Cat, how to translate (in terms of functors) the notion of monomorphicity ?
And since we're at it, how to do the same with the notion of epimorphicity ?
Thanks already for the kind answers !
 A: What it means for a functor $F : \mathcal{C} \to \mathcal{D}$ to be faithful is that, for each $A,B \in \mathrm{ob}(\mathcal{C})$, the function $\mathrm{Hom}_{\mathcal{C}}(A,B) \to \mathrm{Hom}_{\mathcal{D}}(FA, FB)$ is an injection (i.e. a monomorphism in $\mathbf{Set}$). Note that this condition only talks about the morphisms of $\mathcal{D}$, so for instance $F$ could in theory send every object of $\mathcal{C}$ to a single object of $\mathcal{D}$ and still be faithful.
On the other hand, the monomorphisms $F : \mathcal{C} \to \mathcal{D}$ are precisely the embeddings, which are functors that are faithful and injective on objects, so that distinct objects of $\mathcal{C}$ are sent to distinct objects of $\mathcal{D}$. So every monomorphism in $\mathbf{Cat}$ is a faithful functor, but not every faithful functor is a monomorphism.
Epimorphisms in $\mathbf{Cat}$ are a bit trickier to pin down, so I don't think the relationship between epimorphisms and full functors will be quite as easy to describe.
