Lemma 46.3.9. Let $A$ be a ring. Let $\varphi : F \to \underline{M}$ be a map of module-valued functors on $\textit{Alg}_ A$ with $F$ adequate. Then $\mathop{\mathrm{Coker}}(\varphi )$ is adequate.

Proof. By Lemma 46.3.6 we may assume that $F = \bigoplus L_ i$ is a direct sum of linearly adequate functors. Choose exact sequences $0 \to L_ i \to \underline{A^{\oplus n_ i}} \to \underline{A^{\oplus m_ i}}$. For each $i$ choose a map $A^{\oplus n_ i} \to M$ as in Lemma 46.3.8. Consider the diagram

$\xymatrix{ 0 \ar[r] & \bigoplus L_ i \ar[r] \ar[d] & \bigoplus \underline{A^{\oplus n_ i}} \ar[r] \ar[ld] & \bigoplus \underline{A^{\oplus m_ i}} \\ & \underline{M} }$

Consider the $A$-modules

$Q = \mathop{\mathrm{Coker}}(\bigoplus A^{\oplus n_ i} \to M \oplus \bigoplus A^{\oplus m_ i}) \quad \text{and}\quad P = \mathop{\mathrm{Coker}}(\bigoplus A^{\oplus n_ i} \to \bigoplus A^{\oplus m_ i}).$

Then we see that $\mathop{\mathrm{Coker}}(\varphi )$ is isomorphic to the kernel of $\underline{Q} \to \underline{P}$. $\square$

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