Inverse Limit of Complexs and Homology

Question

Let $\mathcal{A}$ be an abelian category and denote by $D(\mathcal{A})$ its derived category. Let $K_n\in D(\mathcal{A})$ be an inverse system of complexes in $\mathcal{A}$ viewed as elements of the derived category. We have a natural map $$ \lim K_n\rightarrow K_n$$ and thus $$H_m(\lim K_n)\rightarrow \lim H_m(K_n).$$ Since $\lim$ is in general not an exact functor, this won't be an isomorphism. Do we know more about this map, for instance is surjective in general? The only case I can deal with is the short exact sequence of complexes $$0\rightarrow A^\bullet \rightarrow B^\bullet \rightarrow C^\bullet\rightarrow 0$$ because that yields a long exact sequence $$\ldots\rightarrow H_m(\lim B^\bullet \rightarrow C^\bullet\leftarrow 0)\rightarrow H_m(B^\bullet)\rightarrow H_m(C^\bullet)\rightarrow \ldots$$ and thus we have a surjection $$H_m(\lim B^\bullet \rightarrow C^\bullet\leftarrow 0)\rightarrow \ker \left( H_m(B^\bullet)\rightarrow H_m(C^\bullet) \right)\cong \lim \left( H_m(B^\bullet)\rightarrow H_m(C^\bullet) \leftarrow H_m(0) \right) .$$ However, I don't know how to generalize this.