Lemma 19.11.4. Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$.

If $|M| \leq \kappa $, then $M$ is the quotient of a direct sum of at most $\kappa $ copies of $U$.

For every cardinal $\kappa $ there exists a set of isomorphism classes of objects $M$ with $|M| \leq \kappa $.

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