For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$.
I know well the plus-construction of sheafification, which is presented in Artin's paper "Grothendieck Topologies", for example.
QUESTION
I have heard that there is another construction of sheafification by using hypercovering, but I can not find good explanation of it.
So, I want some examples to feel the essence of the construction.
In particular, I am interested in the following situation:
Let $C$ be a category with a pretopology $T$.
For an object $X \in C$, denote by $F_{X}$ the presheaf on $C$ represented by $X$ i.e. $F_{X}=Hom_{C}(-,X)$.
Now, let $X_{\bullet} \to X$ be a $T$-hypercover of $X \in C$.
Then, how can the sheafification of $F_{X}$ be written by using $F_{X_{\bullet}}$ ?