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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