Relationship between kernel and homotopy kernel

Question

Let $R$ be a ring, and $C$ the category of $R-$modules (for instance, abelian groups) . We can consider the category $\text{Com}(C)$ of complexes of $R-$modules. We can even consider the category $\text{Kom}(C)$ whose objects are the complexes of $R-$modules and the morphisms are the equivalence class of the morphisms in $\text{Com}(C).$ Consider a morphism in $\text{Com}(C),$ $$\phi^\bullet:M^\bullet\to N^\bullet$$ and let $\eta:K^\bullet\to M^\bullet$ be its kernel. I know that the homotopy category is not abelian, hence in general a morphism in this category cannot have a kernel, but I want to study the case in which $\overline{\phi}$ has its kernel. Suppose then that the kernel of $\overline{\phi}$ exists: is there any relationship with the kernel of the map $\phi?$ I would like to prove that this kernel is given by $\overline{\eta}: K^\bullet\to M^\bullet$ but I’m not sure if it is true and eventually I don’t know how to prove it. Is it true? In this case, how can this be proved? If it is false, is there another relationship?