Let $S^k$ be the presheaf on a space $X$ that assigns to every open set $U$ the abelian group $S^k(U)$ of singular k- cochains on $U$. This is clearly not a sheaf. Consider the sheafification $F^k$ of each $S^k$. These sheaves form an exact resolution of the constant sheaf of integers.
We can take global sections on this sheaf resolution to obtain a cochain complex $F^*(X)$. Does the cohomology of this cochain complex coincides with ordinary singular cohomology?