Definition 12.12.2. Let \mathcal{A}, \mathcal{B} be abelian categories. Let (F^ n, \delta _ F) and (G^ n, \delta _ G) be \delta -functors from \mathcal{A} to \mathcal{B}. A morphism of \delta -functors from F to G is a collection of transformation of functors t^ n : F^ n \to G^ n, n \geq 0 such that for every short exact sequence 0 \to A \to B \to C \to 0 of \mathcal{A} the diagrams
\xymatrix{ F^ n(C) \ar[d]_{t^ n} \ar[rr]_{\delta _{F, A \to B \to C}} & & F^{n + 1}(A) \ar[d]^{t^{n + 1}} \\ G^ n(C) \ar[rr]^{\delta _{G, A \to B \to C}} & & G^{n + 1}(A) }
are commutative.
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