In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble understanding the final step of his proof, and I was wondering if someone here can enlighten me.
Suppose we have an additive left exact functor $F:A\rightarrow B$ between abelian categories. Then let $f:K\rightarrow L$ be a map of chain complexes where K and L consist of objecs that are $F$-acyclic. We have a short exact sequence
$$0\rightarrow K\rightarrow Cyl(f)\rightarrow Cone(f)\rightarrow 0.$$$$0\rightarrow K\rightarrow \operatorname{Cyl}(f)\rightarrow \operatorname{Cone}(f)\rightarrow 0.$$
Recall that the mapping cone of $f$ is acyclic presicely when $f$ is a quasi-isomorphism, and that the mapping cylinder of $f$ is quasi-isomorphic with $L$. Since additive functors preserve mapping cones and mapping cylinders (as they preserve direct sums), and $K$ is $F$-acyclic, we obtain the short exact sequence
$$0\rightarrow F(K)\rightarrow Cyl(F(f))\rightarrow Cone(F(f))\rightarrow 0.$$$$0\rightarrow F(K)\rightarrow\operatorname{Cyl}(F(f))\rightarrow \operatorname{Cone}(F(f))\rightarrow 0.$$
According to the poster of the aforementioned post, the result should now follow. However, I don't see how it does. Most likely we are to prove that Cone(F(f))$\operatorname{Cone}(F(f))$ has trivial homology groups. Can someone help me with this?