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).



How would you say that a small category is embedded into functors from a large $C'$ to abelian groups?
In your example, there likely exists a cardinal $\kappa$ such that $C'$ is [$\kappa$-accessible](http://ncatlab.org/nlab/show/accessible+category), as are the functors $C'\to Ab$ associated to each object of $C$ (i.e., they preserve $\kappa$-filtered colimits). In this case, you can consider $C$ embedded into the category of $\kappa$-accessible functors from $C'$ to $Ab$, which is only "large" rather than "very large". This is equivalent to restricting to the subcategory of $\kappa$-compact objects of $C'$.


