Why is full- & faithful- functor defined in terms of Set properties? 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?

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