The Stacks project

Remark 19.9.4. If $\mathcal{A}$ is a “big” abelian category, i.e., if $\mathcal{A}$ has a class of objects, then Lemma 19.9.2 does not work. In this case, given any set of objects $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ there exists an abelian full subcategory $\mathcal{A}' \subset \mathcal{A}$ such that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ is a set and $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$. Then one can apply Lemma 19.9.2 to $\mathcal{A}'$. One can use this to prove that results depending on a diagram chase hold in $\mathcal{A}$.