I was curious about how to find sufficiently many examples of small non-filtered abelian categories that appear naturally?
I thought of the category of vector spaces with a fixed basis that seems to be non-filtered, but is there any more good examples? (Is my example even correct?)
(For reference: I'm trying to solve the following problem: is it true that colimit of representable functors over comma category is left exact iff the comma category is filtered)
Edit: Is it true that if a small abelian category $\mathcal{A}$ is filtered its category of elements (i.e. comma category $(*\downarrow \mathcal{A}))$ is filtered? (I suppose no, but are there any counterexamples?)