Let $$\dots A_i\stackrel {f_i}\to B_i \stackrel {g_i}\to C_i \stackrel {h_i}\to A_{i+1}\to \dots$$ be a long exact sequence of Abelian groups.
Is it true that if there are maps $k_i:D_i\to E_i$ such that $\ker k_i\simeq \ker g_i$ and $\operatorname{coker} k_i \simeq \operatorname{coker}g_i$, that then a long exact sequence $$\dots A_i\to B_i \stackrel {k_i}\to C_i \to A_{i+1}\to \dots$$ exists?
If yes, what are the maps?