Right exact functor applied to epimorphism of cohomology is still epimorphism?

Question

Let $\mathcal A,\mathcal B$ be abelian categories and $F$ an additive, right exact functor $\mathcal A\rightarrow\mathcal B$. Suppose I have a morphism of chain complexes (in positive degrees) $C^{\,\boldsymbol{\cdot}}\rightarrow D^{\,\boldsymbol{\cdot}}$ in $\mathcal A$ inducing an epimorphism $h^i(C^{\,\boldsymbol{\cdot}})\twoheadrightarrow h^i(D^{\,\boldsymbol{\cdot}})$ in cohomology. Is $h^i(F(C^{\,\boldsymbol{\cdot}}))\rightarrow h^i(F(D^{\,\boldsymbol{\cdot}}))$ epi?

Answer 1

No. Take $C^\bullet$ to be the zero complex, and $D^\bullet$ an acyclic complex such that $F(D^\bullet)$ is not acyclic.