In B. Mitchell. Theory of categories (1965) three properties of abelian categories were introduced.

An abelian category is termed $C_1$ if it is cocomplete with exact coproducts, $C_2$ if it is bicomplete and natural transformation $\eta_{-, X}: \coprod_X (-_x) \to \prod_X (-_x)$ is mono for every set $X$, and $C_3$ if it is cocomplete with exact filtered colimits.

In bicomplete abelian categories the property in the OP is equivalent to $C_2$.

It's obvious that complete $C_3$ category is $C_2$, and $C_2$ always implies $C_1$ (it's in the 3rd chapter of Mitchell's book).

In the paper Exactness of direct limits for abelian categories with an injective cogenerator by L. Positselski and J. Stovicek arXiv one of the results is the following.

Theorem. Let $\mathrm A$ be a complete abelian category with an injective cogenerator $W$. (It is automatically cocomplete by special adjoint functor theorem.) Following conditions are equivalent:

• filtered colimits in $\mathrm A$ are exact;
• $\eta_{W, X}: \coprod_X W \to \prod_X W$ is mono;
• $\operatorname{Hom}(\eta_{W, X}, W): \operatorname{Hom}(W^X, W) \to \operatorname{Hom}(W^{(X)}, W)$ is surjective;
• every morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$;
• summation morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$.

As far as I know, there's no known example without injective cogenerator, $\eta$ being mono and non-exact filtered colimits. If there are no products (and, necessarily, no injective cogenerator), I suspect that this "local finiteness" condition on coproducts behaves quite patologically; it seems very unlikely to imply exactness of filtered colimits. Construction of a cocomplete but not complete abelian category is already quite involved; I haven't checked (yet) whether this construction by J. Rickard serves as a counterexample to the updated question.

In B. Mitchell. Theory of categories (1965) three properties of abelian categories were introduced.

An abelian category is termed $C_1$ if it is cocomplete with exact coproducts, $C_2$ if it is bicomplete and natural transformation $\eta_{-, X}: \coprod_X (-_x) \to \prod_X (-_x)$ is mono for every set $X$, and $C_3$ if it is cocomplete with exact filtered colimits.

In bicomplete abelian categories the property in the OP is equivalent to $C_2$.

It's obvious that complete $C_3$ category is $C_2$, and $C_2$ always implies $C_1$ (it's in the 3rd chapter of Mitchell's book).

In the paper Exactness of direct limits for abelian categories with an injective cogenerator by L. Positselski and J. Stovicek arXiv one of the results is the following.

Theorem. Let $\mathrm A$ be a complete abelian category with an injective cogenerator $W$. (It is automatically cocomplete by special adjoint functor theorem.) Following conditions are equivalent:

• filtered colimits in $\mathrm A$ are exact;
• $\eta_{W, X}: \coprod_X W \to \prod_X W$ is mono;
• every morphism $f: W^{(X)} \to W$ factors through $\eta_{W, X}$;
• summation morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$.

As far as I know, there's no known example without injective cogenerator, $\eta$ being mono and non-exact filtered colimits. If there are no products (and, necessarily, no injective cogenerator), I suspect that this "local finiteness" condition on coproducts behaves quite patologically; it seems very unlikely to imply exactness of filtered colimits. Construction of a cocomplete but not complete abelian category is already quite involved; I haven't checked (yet) whether this construction by J. Rickard serves as a counterexample to the updated question.

In B. Mitchell. Theory of categories (1965) three properties of abelian categories were introduced.

An abelian category is termed $C_1$ if it is cocomplete with exact coproducts, $C_2$ if it is bicomplete and natural transformation $\eta_{-, X}: \coprod_X (-_x) \to \prod_X (-_x)$ is mono for every set $X$, and $C_3$ if it is cocomplete with exact filtered colimits.

In bicomplete abelian categories the property in the OP is equivalent to $C_2$.

It's obvious that complete $C_3$ category is $C_2$, and $C_2$ always implies $C_1$ (it's in the 3rd chapter of Mitchell's book).

In the paper Exactness of direct limits for abelian categories with an injective cogenerator by L. Positselski and J. Stovicek arXiv one of the results is the following.

Theorem. Let $\mathrm A$ be a complete abelian category with an injective cogenerator $W$. (It is automatically cocomplete by special adjoint functor theorem.) Following conditions are equivalent:

• filtered colimits in $\mathrm A$ are exact;
• $\eta_{W, X}: \coprod_X W \to \prod_X W$ is mono;
• $\operatorname{Hom}(\eta_{W, X}, W): \operatorname{Hom}(W^X, W) \to \operatorname{Hom}(W^{(X)}, W)$ is surjective;
• every morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$;
• summation morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$.

As far as I know, there's no known example without injective cogenerator, $\eta$ being mono and non-exact filtered colimits. If there are no products (and, necessarily, no injective cogenerator), I suspect that this "local finiteness" condition on coproducts behaves quite patologically; it seems very unlikely to imply exactness of filtered colimits.

In B. Mitchell. Theory of categories (1965) three properties of abelian categories were introduced.

An abelian category is termed $C_1$ if it is cocomplete with exact coproducts, $C_2$ if it is bicomplete and natural transformation $\eta_{-, X}: \coprod_X (-_x) \to \prod_X (-_x)$ is mono for every set $X$, and $C_3$ if it is cocomplete with exact filtered colimits.

In bicomplete abelian categories the property in the OP is equivalent to $C_2$.

It's obvious that complete $C_3$ category is $C_2$, and $C_2$ always implies $C_1$ (it's in the 3rd chapter of Mitchell's book).

In the paper Exactness of direct limits for abelian categories with an injective cogenerator by L. Positselski and J. Stovicek arXiv one of the results is the following.

Theorem. Let $\mathrm A$ be a complete abelian category with an injective cogenerator $W$. (It is automatically cocomplete by special adjoint functor theorem.) Following conditions are equivalent:

• filtered colimits in $\mathrm A$ are exact;
• $\eta_{W, X}: \coprod_X W \to \prod_X W$ is mono;
• $\operatorname{Hom}(\eta_{W, X}, W): \operatorname{Hom}(W^X, W) \to \operatorname{Hom}(W^{(X)}, W)$ is surjective;
• every morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$;
• summation morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$.

As far as I know, there's no known example without injective cogenerator, $\eta$ being mono and non-exact filtered colimits. If there are no products (and, necessarily, no injective cogenerator), I suspect that this "local finiteness" condition on coproducts behaves quite patologically; it seems very unlikely to imply exactness of filtered colimits. Construction of a cocomplete but not complete abelian category is already quite involved; I haven't checked (yet) whether this construction by J. Rickard serves as a counterexample to the updated question.