Lemma 22.27.17. Let R be a ring. Let F : \mathcal{A} \to \mathcal{B} be a functor between differential graded categories over R satisfying axioms (A), (B), and (C) such that F(x[1]) = F(x)[1]. Then F induces an exact functor K(\mathcal{A}) \to K(\mathcal{B}) of triangulated categories.
Proof. Namely, if x \to y \to z is an admissible short exact sequence in \text{Comp}(\mathcal{A}), then F(x) \to F(y) \to F(z) is an admissible short exact sequence in \text{Comp}(\mathcal{B}). Moreover, the “boundary” morphism \delta = \pi \text{d}(s) : z \to x[1] constructed in Lemma 22.27.1 produces the morphism F(\delta ) : F(z) \to F(x[1]) = F(x)[1] which is equal to the boundary map F(\pi ) \text{d}(F(s)) for the admissible short exact sequence F(x) \to F(y) \to F(z). \square
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