Implication Details

Claim: If a category satisfies CIP and satisfies CSP and is additive and is balanced, then it is trivial.

Proof: We work in a balanced additive category with products and coproducts such that, for every family of objects $(X_i)$, the canonical morphism $$\textstyle \alpha : \coprod_i X_i \to \prod_{i \in I} X_i$$ is a monomorphism and an epimorphism, and therefore an isomorphism. In particular, for every object $A$ the canonical morphism $$\textstyle \alpha : \coprod_{n \geq 1} A \to \prod_{n \geq 1} A$$ is an isomorphism. By this proposition, this implies that $A = 0$.

Remark: This result strengthens the well-known theorem (and is inspired by it) stating that the dual of a Grothendieck abelian category can only be Grothendieck abelian when it is trivial, which appears as exercise B on p. 116 in Freyd's book Abelian categories. This follows from the result we have just proved since a Grothendieck abelian category with products is additive, balanced, and satisfies CIP (by this result).