That categorical definition is for pre-sheaves, the topological definition is for sheaves.
In topological pre-sheaves, a map is surjective if it is epimorphic for each open set $U$ in $X$.
In topological sheaves, however, we instead have to "sheaf-ify" the definition, and we say that the map is "surjective" if the sheaf-ification of the cokernel map is zero.
Basically, in both cases, you have two categories, $\mathcal{Sh}$ and $\mathcal{PSh}$, and in $\mathcal{PSh}$, the "surjective" maps are the ones that are epimorphisms on each $U$, but in the $\mathcal{Sh}$ catageory, you have a more complicated definition of "surjective" (or "epimorphism.")
Consider, instead, two categories, $\mathcal{Ab}$ the category of abelian groups, and $\mathcal{AbTF}$, the full subcategory of "torsion-free" abelian groups - that is, the abelian groups, $A$, where for any $n\in\mathbb Z$ and $a\in A$, $na=0$ iff $n=0$ or $a=0$.
There is the natural inclusion functor $\mathcal{AbTF}\to\mathcal{Ab}$ and a natural adjoint sending $A\to A/N(A)$ where $N(A)$ is the subgroup of nilpotent elements of $A$.
But in $\mathcal{AbTF}$, the "epimorphisms" are not the ones with cokernel (in $\mathcal{Ab}$) $0$, they are the ones with cokerkels which are nilpotent. So, for example, in $\mathcal{Ab}$, the morphism $\mathbb Z\to\mathbb Z$ sending $x\to 2x$ is not an epimorphism, that same map, when considered as a map in $\mathcal{AbTF}$, is an epimorphism.
So consider the "sheafification" functor $\mathcal{PSh}\to \mathcal{Sh}$ to be much like the functor $\mathcal{Ab}\to\mathcal{AbTF}$.
(I believe, but don't quote me, that $f:A\to B$ in $\mathcal{AbTF}$ is an epimorphism if and only if $f\otimes \mathbb Q:A\otimes \mathbb Q\to B\otimes\mathbb Q$ is an epimorphism in $\mathcal{Ab}$.)
To construct free resolutions what you want is a functor $G : A \to \text{Set}$ ($A$ an abelian category) which
- has a left adjoint $F : \text{Set} \to A$ and
- preserves epimorphisms.
These conditions are satisfied, for example, by the usual forgetful functor from $R$-modules to sets. Note that we do not need to assume faithfulness. Now we can prove the following:
Lemma ("free objects are projective"): With the above hypotheses, $F$ preserves projective objects.
Here we need to define "projective" to mean $\text{Hom}(P, -)$ preserves epimorphisms, in order to correctly apply to nonabelian categories. Now the proof is very short: if $P \in \text{Set}$ is a projective object (every set is projective assuming the axiom of choice), then
$$\text{Hom}(F(P), -) \cong \text{Hom}(P, G(-))$$
so $\text{Hom}(F(P), -)$ is a composite of two functors which preserve epimorphisms and hence preserves epimorphisms.
Best Answer
No, that does not follow from the definition as a contravariant functor, and it is incorrect to add it to the definition of a presheaf. I've never seen a source that does this, and it would be interesting if you would include a citation. The condition does follow from the definition of a sheaf, though.
EDIT: There is no more reason to make this requirement if the values are in an abelian category than if the values are in sets. A presheaf is simply a contravariant functor, full stop.