Let $R$ be a commutative unitary ring and $\mathrm{Mod}_R$ be the category of $R$-modules and $C := \mathrm{Ch}_{\geq 0}(\mathrm{Mod}_R)$ be the category of chains of $R$-modules which are zero in negative degree. Then we have a functor $H^n: C \to \mathrm{Mod}_R, M^{\bullet} \mapsto H^0(M^{\bullet})$. Since $H^n$ vanishes on exact sequences, $H^n$ induces a functors in the derived category: $H^n_1: D(\mathrm{Mod}_R) \to \mathrm{Mod}_R$.
On the other hand, I have the functor $H^0: C \to \mathrm{Mod}_R$ which induces a functor $D(H^0): D(\mathrm{Mod}_R) \to D(\mathrm{Mod}_R)$. I want to show that $H^n_1 = H^n_1 \circ D(H^0)$. Does anybody know how to prove this?