How would you say that a small additive category $C$ embedds (contravariantly) into the category of exact functors from a 'large' abelian $C'$ into abelian groups (this is something like Yoneda's embedding, but $C$ does not map canonically into $C'$)? My problem is that I do not want to consider all functors from $C'$ into abelian groups since this functor category it 'very large' (and I do not want to consider a 'larger universe'). Certainly, I can try to consider a limit of the corresponding functors from small subcategories of $C'$; yet is there a better way to deal with this matters?

Upd. Actually, my $C'$ is just isomorphic to the category of additive functors from $C^{op}$ to abelian groups (though this is not the way how it is defined).