I've encountered The following categorical characterization of finitely generated modules:
A $R$-module $M$ is finitely generated iff it satifies one of the following properties:
a): for any family of $R$-module $\{U_i\}_{i\in\mathcal I}$ and any epimorphism $f:\bigoplus_{i\in\mathcal I} U_i\twoheadrightarrow M$, there exists a finite subset $\mathcal F$ of $\mathcal I$ such that the restriction of $f$ on $\bigoplus_{i\in\mathcal F} U_i$ is also epic.
b): for any category $\mathscr I$ representing a directed partially ordered set and any functor $F:\mathscr I\to R\text -\mathsf{Mod}$ such that every arrow of $\mathscr I$ is mapped to an injection via $F$, then the canonical homomorphism $\varphi:\varinjlim\text{Hom}(M,-)\circ F\to\text{Hom}(M,\varinjlim F)$ is epic in $\mathsf{Ab}$.
My question is:
If $R\text -\mathsf{Mod}$ is replaced by an arbitrary cocomplete abelian category, are characterization a) and b) remain equivalent?
I've already proved that b) always implies a).
The two are not equivalent in the opposite category of the category of vector spaces over a field $k$, since a one-dimensional vector space $k$ satisfies (a) but not (b).
It's amazing how often opposite categories of module categories work as counterexamples!