Lemma 46.3.11. Let $A$ be a ring. Let $\varphi : F \to G$ be a map of adequate functors on $\textit{Alg}_ A$. Then $\mathop{\mathrm{Ker}}(\varphi )$ is adequate.

Proof. Choose an injection $F \to \underline{M}$ and an injection $G \to \underline{N}$. Denote $F \to \underline{M \oplus N}$ the diagonal map so that

$\xymatrix{ F \ar[d] \ar[r] & G \ar[d] \\ \underline{M \oplus N} \ar[r] & \underline{N} }$

commutes. By Lemma 46.3.10 we can find a module map $M \oplus N \to K$ such that $F$ is the kernel of $\underline{M \oplus N} \to \underline{K}$. Then $\mathop{\mathrm{Ker}}(\varphi )$ is the kernel of $\underline{M \oplus N} \to \underline{K \oplus N}$. $\square$

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