This question is motivated by this. I will use the notations of my answer to this.
We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$).
Let $\mathcal A$ be an abelian category which may not be locally small. Let $V$ be a finite subset of Ob($\mathcal A$). Let $T$ be a finite subset of Mor($\mathcal A$) such that both dom($f$) and codom($f$) belong to $V$ for each $f \in T$. Does there exist an abelian subcategory $\mathcal B$ of $\mathcal A$ satisfying the following conditions? If yes, how would you prove it?
(1) $\mathcal B$ is small.
(2) $V \subset$ Ob($\mathcal B$) and $T \subset$ Mor($\mathcal B$).
(3) The canonical functor $\mathcal B \rightarrow \mathcal A$ is exact.
If the answer is affirmative, by Mitchell's embedding theorem, you can consider the diagram ($V, T$) as that of the category of modules over a ring. Hence you can use diagram chasing on it.
