Hypercohomology and acyclicity of direct images

Question

Let $f: X\rightarrow Y$, $g:Y \rightarrow Z$ be morphisms of topological spaces and let $K^{.}$ be an injective object in the category of complexes of abelian sheaves on $X$. Write $\mathbb R^0f_* K^.$ for the zeroth hyperderived image of this complex under $f$.

Then, why is $\mathbb R^0f_* K^.$ acyclic for the functor $g_*$ ?