TITLE: Abelian category which is not well-powered
QUESTION [10 upvotes]: Can you give an example of an abelian category which is not well-powered? If not, maybe you can give any reason why there are such abelian categories?

REPLY [7 votes]: Here's a simpler, but less consequential, example.
Take the category of "eventually constant" functors from ordinals (considered as a category with a single morphism $\alpha\to\beta$ when $\alpha\leq\beta$) to abelian groups, meaning functors $F$ for which there is some ordinal $\alpha$ such that $F(\beta)\to F(\gamma)$ is an isomorphism for all $\alpha\leq\beta\leq\gamma$. 
The "eventually constant" condition ensures that this is a locally small category. It is not well-powered, since for any ordinal $\alpha$, the constant functor taking value $\mathbb{Z}$ has a subfunctor $F_\alpha$ with
$$F_\alpha(\beta)=\begin{cases}0&\text{if $\beta<\alpha$}\\
\mathbb{Z}&\text{if $\beta\geq\alpha$.}
\end{cases}$$