The Stacks project

Proof. Let $(N, \varphi )$ be a descent datum for $R \to A$. Set $N' = R' \otimes _ R N = A' \otimes _ A N$, and denote $\varphi ' = \text{id}_{R'} \otimes \varphi $ the base change of the descent datum $\varphi $. Then $(N', \varphi ')$ is a descent datum for $R' \to A'$ and $H^0(s(N'_\bullet )) = R' \otimes _ R H^0(s(N_\bullet ))$. Moreover, the map $A' \otimes _{R'} H^0(s(N'_\bullet )) \to N'$ is identified with the base change of the $A$-module map $A \otimes _ R H^0(s(N)) \to N$ via the faithfully flat map $A \to A'$. Hence we conclude by Lemma 35.3.7. $\square$