$\newcommand{\im}{\operatorname{im}}$I am losing my sanity over this diagram chase.
Given an abelian category and a cochain complex $A$, I am trying to prove that $\ker(A^i/\im d^{i+1} \to \ker d^{i-1})$ is isomorphic to $H^i(A) = \operatorname{coker}(\im d^{i+1}\to \ker d^i)$.
This is apparently how you get the long exact sequence from the snake lemma if you apply it to the correct diagram but I cannot confirm this without going into an element-wise proof. All I can get is a morphism from $H^i(A)$ to the other object but not the other way around and this is the only way I know how to show that two objects are isomorphic.
Am I supposed to use the fact that coimages are isomorphic to images in abelian categories? Because that did not help much either. A tip would be appreciated if it's possible.