Vanishing of the Ext groups $\mathrm{Ext}^2(H^nX,-)$ In the book Interactions Between Homotopy Theory and Algebra, there is the following introduction to hereditary categories. I wonder why the vanishing of $\mathrm{Ext}_A^2 (H^nX,-)$ implies the two rows exact sequences as follows. Can anyone explain this, please?

 A: By definition of $Ext$ (or so I hope), $E^n$ (and the topmost row) is defined by a class $x \in \mathrm{Ext}^1(H^nX,X^{n-1})$. The constraint is that the image of this $x$ under the map $d_{n-1}: X^{n-1} \rightarrow \mathrm{Im}\,d^{n-1}$ is the class corresponding to the extension $\ker{d^n}$ in $\mathrm{Ext}^1(H^nX,\mathrm{Im}\,d^{n-1})$.
In other words, we want to show that if $B \rightarrow C$ is an epimorphism in the category, and $A$ is another object, $\mathrm{Ext}^1(A,B) \rightarrow \mathrm{Ext}^1(A,C)$ is surjective (that’s more than enough, but such is often the case in homological algebra): to see that this is enough, apply it to $A=H^nX$ and $B \rightarrow C$ being $d_{n-1}: X^{n-1} \rightarrow \mathrm{Im}\,d^{n-1}$.
But now, we have the long exact sequence, if $K$ is the kernel of $B \rightarrow C$: $\mathrm{Ext}^1(A,B) \rightarrow \mathrm{Ext}^1(A,C) \rightarrow \mathrm{Ext}^2(A,K)$. But we have assumed that $\mathrm{Ext}^2$ vanished in our category! So $\mathrm{Ext}^1(A,B) \rightarrow \mathrm{Ext}^1(A,C)$ is surjective and we are done.
