Lemma 21.26.1. In the situation above, choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map for one of the four arrows we get maps of complexes
\[ \mathcal{I}^\bullet (X) \xrightarrow {\alpha } \mathcal{I}^\bullet (Z) \oplus \mathcal{I}^\bullet (Y) \xrightarrow {\beta } \mathcal{I}^\bullet (E) \]
with $\beta \circ \alpha = 0$. Thus a canonical map
\[ c^ K_{X, Z, Y, E} : \mathcal{I}^\bullet (X) \longrightarrow C(\beta )^\bullet [-1] \]
This map is canonical in the sense that a different choice of K-injective complex representing $K$ determines an isomorphic arrow in the derived category of abelian groups. If $c^ K_{X, Z, Y, E}$ is an isomorphism, then using its inverse we obtain a canonical distinguished triangle
\[ R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K) \to R\Gamma (X, K)[1] \]
All of these constructions are functorial in $K$.
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