The two Ext long exact sequences agree under balance

For $0\to N^{\mathrm{\prime }}\to N\to N^{\mathrm{\prime }\mathrm{\prime }}\to 0$, fix $P\to M$ and choose an injective horseshoe $0\to I^{\mathrm{\prime }}\to I\to I^{\mathrm{\prime }\mathrm{\prime }}\to 0$. There are three short exact sequences of cochain complexes: $\mathrm{Hom}(P,N^{\ast })$, $\mathrm{Hom}(M,I^{\ast })$, and $T(P,I^{\ast })$, where $\ast$ denotes the three terms of the short exact sequence, not cochain degree. The first is exact by projectivity of each $P_p$; the second and third are exact because the horseshoe is split in each degree, and total diagonals are finite. The coaugmentations and augmentation give two morphisms of short exact sequences from the edge sequences to the total sequence.

For $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$, fix $N\to I$ and choose a projective horseshoe $0\to P^{\mathrm{\prime }}\to P\to P^{\mathrm{\prime }\mathrm{\prime }}\to 0$. The three short exact sequences are $\mathrm{Hom}(P^{\ast },N)$, $\mathrm{Hom}(M^{\ast },I)$, and $T(P^{\ast },I)$, all ordered with double-prime first and prime last. The first and third are exact by the degreewise splitting; the second is exact by injectivity of each $I^q$. Again the augmentation and coaugmentation give morphisms from both edge sequences to the total sequence.

In each case let $a$ be the projective-edge map to the total and $b$ the injective-edge map. By [F3], $H(a)$ and $H(b)$ commute with the connecting maps of their respective sequences and the total sequence. They are isomorphisms by [F1]. Hence $\beta =H(b{)}^{-1}H(a)$ also commutes with connecting maps. These are the actual balance maps, not merely some degreewise natural isomorphism. The total differential is $h+(-1{)}^{p}v$; both edge maps are cochain maps with the page's unsigned Hom differentials, so these are commuting squares with no additional sign.

The horseshoe middle resolutions may differ from the fixed ones. Transport their cohomology to the supplied data by comparison isomorphisms. By [F3] this is exactly how the derived connecting maps are defined, and [F1] makes balance commute with these comparisons. Therefore the squares proved in step 2.1 hold for the supplied resolutions as well. In degree zero both augmentations identify the common cocycles with Hom, giving its identity identification.