I could name on the spot a bunch of abelian categories satisfying AB5 but I cannot think of any that satisfies AB5\*. That is, it should have all limits and the cofiltered limits are exact. Is there any example that I should have in mind?



Abelian categories satisfying AB5*
The snarky response would be "the opposite category of any of the categories you could name on the spot". The less-snarky response is to observe that some of these are quite natural. For example, $\mathrm{Vect}^{\mathrm{op}}\_{\mathbb{F}\_p}$ agrees with the category of profinite $\mathbb{F}\_p$ vector spaces, since $\mathrm{Vect}\_{\mathbb{F}\_p}$ is the $\mathrm{Ind}$-category of finite $\mathbb{F}\_p$ vector spaces, finite vector spaces are self-dual, and profinite $\mathbb{F}\_p$ vector spaces are the $\mathrm{Pro}$-category of finite vector spaces.


