Let $\mathcal{A}$ be an Abelian category, $X$ be a complex, $F$ be a contravariant exact functor. I am wondering whether F preserves the homology of X, that means whether $H^{i}(FX)=F(H^{-i}(X)),\ \forall i$? (Obviously, this is true for covariant functors)



homology under exact functors
Yes. There are two ways to describe $H^iX$: it is the cokernel of $X^{i-1}\to ker(X^i\to X^{i+1})$, and it is also the kernel of $coker(X^{i-1}\to X^i)\to X^{i+1}$. From the first of these it is clear that $F(H^iX)$ is the kernel of $coker(FX^{i+1}\to FX^i)\to FX^{i-1}$. (And from the second it is clear that $F(H^iX)$ is the cokernel of $FX^{i+1}\to ker(FX^i\to FX^{i-1})$.)


