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Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?

I suspect that in certain cases one can take a functor represented by an injective cogenerator in the category $\operatorname{Ind}-A$, but I don't understand when this works. Do exact $\alpha$-filtered colimits in $A$ (where $\alpha$ is a regular cardinal) help (say, if $A$ contains a generator)?

Upd. As kindly note by Qiaochu Yuan below, my assumption is equivalent to having an injective cogenerator in $A$. So, I would be grateful for the relation of the latter assumption to the exactness of $\alpha$-filtered colimits in $A$ (say, along with the assumption that there exists a set of $A_i\in A$ such that $A$ equals its smallest abelian subcategory containing $A_i$ and closed with respect to coproducts).

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?

I suspect that in certain cases one can take a functor represented by an injective cogenerator in the category $\operatorname{Ind}-A$, but I don't understand when this works. Do exact $\alpha$-filtered colimits in $A$ (where $\alpha$ is a regular cardinal) help (say, if $A$ contains a generator)?

Upd. Sorry; I mistunderstood Qiaochu Yuan's answer below. His answer only treats the existence of functors that satisfy certain non-trivial additional conditions, whereas any functor from $A^{op}$ into abelian groups that respects products will be sufficient for my purposes. So, do there exist any re-formulations of this existence, or sufficient assumptions that imply it?

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?

I suspect that in certain cases one can take a functor represented by an injective cogenerator in the category $\operatorname{Ind}-A$, but I don't understand when this works. Do exact $\alpha$-filtered colimits in $A$ (where $\alpha$ is a regular cardinal) help (say, if $A$ contains a generator)?

Upd. As kindly note by Qiaochu Yuan below, my assumption is equivalent to having an injective cogenerator in $A$. So, I would be grateful for the relation of the latter assumption to the exactness $\alpha$-filtered colimits in $A$ (say, along with the assumption that there exists a set of $A_i\in A$ such that $A$ equals its smallest abelian subcategory containing $A_i$ and closed with respect to coproducts).

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?

I suspect that in certain cases one can take a functor represented by an injective cogenerator in the category $\operatorname{Ind}-A$, but I don't understand when this works. Do exact $\alpha$-filtered colimits in $A$ (where $\alpha$ is a regular cardinal) help (say, if $A$ contains a generator)?

Upd. As kindly note by Qiaochu Yuan below, my assumption is equivalent to having an injective cogenerator in $A$. So, I would be grateful for the relation of the latter assumption to the exactness of $\alpha$-filtered colimits in $A$ (say, along with the assumption that there exists a set of $A_i\in A$ such that $A$ equals its smallest abelian subcategory containing $A_i$ and closed with respect to coproducts).

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?

I suspect that in certain cases one can take a functor represented by an injective cogenerator in the category $\operatorname{Ind}-A$, but I don't understand when this works. Do exact $\alpha$-filtered colimits in $A$ (where $\alpha$ is a regular cardinal) help (say, if $A$ contains a generator)?

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?

I suspect that in certain cases one can take a functor represented by an injective cogenerator in the category $\operatorname{Ind}-A$, but I don't understand when this works. Do exact $\alpha$-filtered colimits in $A$ (where $\alpha$ is a regular cardinal) help (say, if $A$ contains a generator)?

Upd. As kindly note by Qiaochu Yuan below, my assumption is equivalent to having an injective cogenerator in $A$. So, I would be grateful for the relation of the latter assumption to the exactness $\alpha$-filtered colimits in $A$ (say, along with the assumption that there exists a set of $A_i\in A$ such that $A$ equals its smallest abelian subcategory containing $A_i$ and closed with respect to coproducts).