How to define homology functor on chain maps

Question

Let $\mathcal{A}$ be an abelian category. I want to define the homology functor $H$ from the category $\operatorname{Ch}(\mathcal{A})$ of chain complexes in $\mathcal{A}$ to itself. The following

How to define Homology Functor in an arbitrary Abelian Category?

answers my question for objects. But what about morphisms? If $f:C\longrightarrow D.$ is a chain map, how can be defined $H(f)$?

Answer 1

Define $H_n(f)$ such that the composite $Ker(d^C_{n+1})\rightarrow H_n(C)\rightarrow H_{n}(D)$ coincides with $Ker(d^C_{n+1})\rightarrow Ker(d^D_{n+1})\rightarrow H_n(D)$ (first arrow induced by $f$) by using the universal properties of Ker/Coker.