Lemma 13.9.10. Let \mathcal{A} be an additive category. Let 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 be termwise split exact sequences as in Definition 13.9.9. Let (\pi ')^ n, (s')^ n be a second collection of splittings. Denote \delta ' : C^\bullet \longrightarrow A^\bullet [1] the morphism associated to this second set of splittings. Then
(1, 1, 1) : (A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta ) \longrightarrow (A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta ')
is an isomorphism of triangles in K(\mathcal{A}).
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