Wikipedia entry or Roman's "Lattices and Ordered Sets" p.286, or Bergman's General Algebra and Universal Constructions, p.177 and in fact every definition of full and/or faithful functor is defined in terms of the Set-theoretical properties: surjective and injective on (compatible) arrows.
Why aren't full-/faithful- defined in terms of epic and monic, in other words, in terms of algebraic invertibility or cancellation properties, eg if it is required to consider not a set of arrows but a topology or order (or any other category) of them.
Is this an historical accident awaiting suitable generalization, or is there some fundamental reason why Set seems to always lurk in the background?
Best Answer
Full and Faithful can be easily defined in general with no reference to Set. Just state:
Faithful functor F
$\forall (f,g: A \to B)$: $Ff = Fg$ implies $f = g$
Full functor F
$\forall (h: FA\to FB)$ $\exists (f: A \to B):Ff = h$
Just FOL, no set theory or category of sets.
You can find this in CWM chapter 1 section Functors