Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} \longrightarrow \mathbf{Mod}_R$ in a category of $R$-modules, for some ring $R$.

Now, $V$ being exact is the same as saying that it preserves all *finite* limits and colimits.

I would be glad to know if Mitchell's embedding theorem could be improved in order to have that $V$ preserves also:

(a) arbitrary products, and (b) filtered colimits.

Or, alternatively,

(c) injective objects.

Or, which conditions on the abelian category ${\cal A}$ would guarantee (a) and (b), or (c)? Are there any results in these directions?

The reason behind my question is the following: I realized that my answer to my previous question http://mathoverflow.net/questions/29883/vanishing-theorems is wrong: sheaf cohomology is not defined uniquely in terms of exact sequences, so the fact that $V$ is exact doesn't guarantee that $H^n(X; {\cal F}) = H^n(X; V({\cal F}))$ as I claimed. But, if I had (a) and (b), I could say that $V$ preserves Godement resolutions. And if I had (c), $V$ would preserve injective resolutions.