Global Section Functor Existence for Pre(Co)Sheaves?

Question

I currently have a co-presheaf F : Top -> Vect. I don't know how to cosheafify it. It is well-known that the Global section functor for sheaves is a covariant left exact functor (right exact for cosheaves). Is this also defined for a (co) presheaf?

Answer 1

A (co)presheaf on a space $X$ is the same thing as a (co)sheaf on the space $Y$ whose underlying set is the set of all open subsets of $X$, with a basis for the open subsets of $Y$ being the sets of the form $d(U)=\{V\in Y:V\subseteq U\}$ for each open subset $U\subseteq X$. In $Y$, no basic open set $d(U)$ can be written as a union of strictly smaller basic open sets (since none of them can contain the point $U$), and it follows that any functor on the poset of basic open sets extends uniquely to a (co)sheaf. But such a functor is just a (co)presheaf on $X$, so you get an equivalence between the category of (co)presheaves on $X$ and the category of (co)sheaves on $Y$.

So if you know that the global sections functor is left (right) exact on (co)sheaves on any space, you automatically get the same result for (co)presheaves on any space.

(Incidentally, I have not thought about it myself much, but according to the answers to this question on MO, describing the cosheafification of copresheaves (or even proving that it exists) seems to be quite nontrivial.)