Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 6 / c / Solution

The commutative diagram of short exact sequences of complexes induces a commutative diagram between the two long exact homology sequences from part (b). If any two vertical chain maps induce isomorphisms in every degree, then in each five-term window four of the five vertical homology maps are isomorphisms. The Five lemma makes the remaining map an isomorphism. Rotating the window handles each of the three possible missing columns, proving the two-out-of-three assertion.