# How much can the Freyd-Mitchell embedding theorem be improved?

Can the embedding theorem be improved to preserve injective objects or even better be part of an adjoint pair? Or if that's not possible are there stronger conditions (AB5, enough injectives, etc) on the abelian category in question that would suffice?

Being explicit, if $$\cal{A}$$ is a small abelian category is there an exact embedding $$F:\cal{A} \rightarrow \mathbf{MOD}_R$$ that preserves injective objects?

• I'm not sure about injectives-I'm skeptical but can't see how to construct a counterexample yet. But you certainly can't get an adjunction, since $\mathcal A$ is small. Any adjunction with image including a nonzero object $x$ of $\mathcal A$ would imply the existence of either coproducts or products of arbitrarily many copies of $x$, which would have an unbounded number of endomorphisms so $\mathcal A$ couldn't be small. – Kevin Carlson Apr 24 at 14:19