Lemma 35.4.17. For $(M, \theta ) \in DD_{S/R}$, the diagram

35.4.17.1
$$\label{descent-equation-coequalizer-CM} \xymatrix@C=8pc{ C(M \otimes _{S, \delta _{12}^1} S_3) \ar@<1ex>[r]^{C((\theta \otimes \delta _2^2) \circ (1_ M \otimes \delta ^2_0))} \ar@<-1ex>[r]_{C(1_{M \otimes S_2} \otimes \delta ^2_1)} & C(M \otimes _{S, \delta _1^1} S_2 ) \ar[r]^{C(\theta \circ (1_ M \otimes \delta _0^1))} & C(M). }$$

obtained by applying $C$ to (35.4.16.1) is a split coequalizer.

Proof. Omitted. $\square$

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