Lemma 84.14.6. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume

1. $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,

2. $K$ is cartesian,

3. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i - 1], K'_ i) = 0$ for $i > 1$.

Then any map $\{ K_ n \to K'_ n\}$ between the associated simplicial systems of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{O})$.

Proof. The proof is exactly the same as the proof of Lemma 84.13.6 except using Lemma 84.14.4 instead of Lemma 84.13.4. $\square$

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