Lemma 19.11.1. Let $\mathcal{A}$ be an abelian category with a generator $U$ and $X$ and object of $\mathcal{A}$. If $\kappa $ is the cardinality of $\mathop{\mathrm{Mor}}\nolimits (U, X)$ then

There does not exist a strictly increasing (or strictly decreasing) chain of subobjects of $X$ indexed by a cardinal bigger than $\kappa $.

If $\alpha $ is an ordinal of cofinality $> \kappa $ then any increasing (or decreasing) sequence of subobjects of $X$ indexed by $\alpha $ is eventually constant.

The cardinality of the set of subobjects of $X$ is $\leq 2^\kappa $.

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