Lemma 46.3.20. Let $A \to A'$ be a ring map and let $F$ be a module-valued functor on $\textit{Alg}_ A$ such that

the restriction $F'$ of $F$ to the category of $A'$-algebras is adequate, and

for any $A$-algebra $B$ the sequence

\[ 0 \to F(B) \to F(B \otimes _ A A') \to F(B \otimes _ A A' \otimes _ A A') \]is exact.

Then $F$ is adequate.

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