Functor between chain complexes $\overset{?}{\implies}$ preserves zero-objects?

Question

Let $\mathsf A,\mathsf B$ be two abelian categories. Any functor $F:A\longrightarrow B$ which preserves zero objects lifts to a functor $\mathsf{Ch}(A)\longrightarrow \mathsf{Ch}(B)$. Furthermore, if it is additive, it lifts to a 2-functor since it preserves chain homotopy.

What about the converse direction? Does a functor $\mathsf{Ch}(\mathsf A)\longrightarrow \mathsf{Ch}(\mathsf B)$ descend to a pointed functor $\mathsf A\rightarrow \mathsf B$? Does a 2-functor desecnd to an additive one?

Answer 1

No. Pick a non-zero object $X$ of $\mathsf{Ch}(\mathsf B)$, and take the $2$-functor sending every object of $\mathsf{Ch}(\mathsf A)$ to $X$, every map to the identity map of $X$, and every homotopy to the trivial homotopy.