Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ such that $U\cup V = X$, we have that
$$ F(X) \stackrel{j_U^* \oplus j_V^*}{\longrightarrow} F(U) \oplus F(V) \stackrel{i_U - i_V}{\longrightarrow} F(U\cap V) \longrightarrow 0$$
is right exact (here, $j_U, j_V$ are the inclusions of $U$ and $V$ into $X$ and $i_U, i_V$ are the inclusions of $U\cap V$ into $X$.

Under what circumstances can we construct functors $F^{-i}$ together with boundary maps that continue the sequence to the left? Is there some general construction?

Of course, this is something like a "left derived functor", but $\mathrm{Man}$ or $\mathrm{Top}$ are not abelian categories, so this doesn't make sense.