Lemma 36.9.3. In Situation 36.9.1. Let $M^\bullet $ be a complex of $A$-modules and denote $\mathcal{F}^\bullet $ the associated complex of $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism of complexes

functorial in $M^\bullet $.

Lemma 36.9.3. In Situation 36.9.1. Let $M^\bullet $ be a complex of $A$-modules and denote $\mathcal{F}^\bullet $ the associated complex of $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism of complexes

\[ \mathop{\mathrm{colim}}\nolimits _ e \text{Tot}(\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet )) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \]

functorial in $M^\bullet $.

**Proof.**
Immediate from Lemma 36.9.2 and our conventions for taking associated total complexes.
$\square$

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