# Can one construct Freyd-Mitchell's embeddings that respect sheafifications?

For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is an isomorphism for a certain morpism of 'spaces' $f$. The results is known when $A$ is the category of abelian groups (actually, this is Voevodsky's theorem that Nisnevich sheafifications preserve homotopy invariance of sheaves with transfers). How can I deduce the statement for a general $A$? I can certainly assume that $A$ is small here; so there exists an exact conservative functor from $A$ to abelian groups (by the Freyd-Mitchell's embedding theorem). Yet is it possible to assume that this functor respects sheafifications? Note that sheafifications involve (infinite) small filtered inductive limits.

I have some ideas how this can be proved using https://mathoverflow.net/a/47762/2191; yet I would certainly prefer to find a reference for this statement (or for some related ones).

• How do you sheafify presheaves with values in $A$ if $A$ doesn't have small filtered colimits? (A small abelian category that has small filtered colimits is necessarily trivial.) – Zhen Lin Sep 20 '13 at 10:09
• Well, I do not actually need 'very large' small colimits; everything is of bounded cardinality. If I fix $P$, then I can assume that A is small. – Mikhail Bondarko Sep 20 '13 at 10:13
• Very related questions (but also unsolved): mathoverflow.net/questions/29883 and mathoverflow.net/questions/32173 – Martin Brandenburg Sep 21 '13 at 8:24
• Yes; yet I only need limits of bounded cardinality. – Mikhail Bondarko Sep 21 '13 at 9:01