# A finite diagram in an abelian category which may not be locally small

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.

-
Interesting question. I think the answer doesn't change if you replace "locally small" with "small", but I could be wrong. Perhaps you should ask on MO? – Zhen Lin Jul 7 '12 at 4:13
Thanks. I replaced "locally small" with "small". – Makoto Kato Jul 7 '12 at 4:19
I posted this question in MO. mathoverflow.net/questions/102616/… – Makoto Kato Jul 19 '12 at 5:19