The cokernel of a map of adequate functors on the category of algebras over a ring is adequate.

Lemma 46.3.10. Let $A$ be a ring. Let $\varphi : F \to G$ be a map of adequate functors on $\textit{Alg}_ A$. Then $\mathop{\mathrm{Coker}}(\varphi )$ is adequate.

Proof. Choose an injection $G \to \underline{M}$. Then we have an injection $G/F \to \underline{M}/F$. By Lemma 46.3.9 we see that $\underline{M}/F$ is adequate, hence we can find an injection $\underline{M}/F \to \underline{N}$. Composing we obtain an injection $G/F \to \underline{N}$. By Lemma 46.3.9 the cokernel of the induced map $G \to \underline{N}$ is adequate hence we can find an injection $\underline{N}/G \to \underline{K}$. Then $0 \to G/F \to \underline{N} \to \underline{K}$ is exact and we win. $\square$

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Suggested slogan: The cokernel of a map of adequate functors on the category of algebras over a ring is adequate.

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