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

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

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

Proof. For (1) choose for every proper subobject $M' \subset M$ a morphism $\varphi _{M'} : U \to M$ whose image is not contained in $M'$. Then $\bigoplus _{M' \subset M} \varphi _{M'} : \bigoplus _{M' \subset N} U \to M$ is surjective. It is clear that (1) implies (2). $\square$

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