Suppose that we are given a nice space $X$ and a sheaf of abelian groups $F$ on $X$. Fix an integer $n$. Then We have a contravariant functor from nice spaces over $X$ to abelian groups; Namely, to a space $f: Y \to X$ we associate the abelian group $H^n (Y, f^{*}F)$ (Sheaf cohomology).

If $X$ is a point, Then this functor is represented, in the homotopy category, by the Eilenberg-Maclane space $K(F,n)$.

My questions are:

1) Can one formalize what will it mean for our functor to be representable, "homotopically"? I am not very sure, but I suspect that the most naive definition of homotopy category of spaces over $X$, and the requirement that the functor is representable in this category, are not right (and I did not check that the functor actually factors to this "homotopy" category).

2) Is it representable?