Section outline
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I showed that a morphism in an abelian category induces a morphism of projective resolutions which is unique up to homotopy equivalence. This is Gelfand–Manin III.1.3–1.4.
Then we constructed mapping cones and distinguished triangles in the homotopy category. We showed that distinguished triangles satisfies the five axioms TR0–TR5. You can find this in Kashiwara–Schapira, Sheaves on Manifolds, I.1.4.