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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