The Stacks project

Lemma 46.3.12. Let $A$ be a ring. An arbitrary direct sum of adequate functors on $\textit{Alg}_ A$ is adequate. A colimit of adequate functors is adequate.

Proof. The statement on direct sums is immediate. A general colimit can be written as a kernel of a map between direct sums, see Categories, Lemma 4.14.12. Hence this follows from Lemma 46.3.11. $\square$

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