Lemma 13.12.1. Let \mathcal{A} be an abelian category. The functor \text{Comp}(\mathcal{A}) \to D(\mathcal{A}) defined has the natural structure of a \delta -functor, with
\delta _{A^\bullet \to B^\bullet \to C^\bullet } = - p \circ q^{-1}
with p and q as explained above. The same construction turns the functors \text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A}), \text{Comp}^{-}(\mathcal{A}) \to D^{-}(\mathcal{A}), and \text{Comp}^ b(\mathcal{A}) \to D^ b(\mathcal{A}) into \delta -functors.
Proof.
We have already seen that this choice leads to a distinguished triangle whenever given a short exact sequence of complexes. We have to show that given a commutative diagram
\xymatrix{ 0 \ar[r] & A^\bullet \ar[r]_ a \ar[d]_ f & B^\bullet \ar[r]_ b \ar[d]_ g & C^\bullet \ar[r] \ar[d]_ h & 0 \\ 0 \ar[r] & (A')^\bullet \ar[r]^{a'} & (B')^\bullet \ar[r]^{b'} & (C')^\bullet \ar[r] & 0 }
we get the desired commutative diagram of Definition 13.3.6 (2). By Lemma 13.9.2 the pair (f, g) induces a canonical morphism c : C(a)^\bullet \to C(a')^\bullet . It is a simple computation to show that q' \circ c = h \circ q and f[1] \circ p = p' \circ c. From this the result follows directly.
\square
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