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).