In an additive category, What are sufficient conditions for the canonical morphism from the coproduct to the product of arbitary collection of objects to be monic (when they both exist)? the conditions can be either on the category or on the objects.
(Helpful related question: proving this statement in abelian category without using the Freyd-Mitchell embedding theorem.) -irrelevant in view of Qiaochu's answer
Since you mentioned locally presentable categories, I'll give one sufficient condition involving that.
Let $\mathcal{A}$ be a locally finitely presentable additive category. Then the canonical morphism $\sum_{i \in I} A_i \to \prod_{i \in I} A_i$ is a monomorphism. This is true when $I$ is finite because $\mathcal{A}$ is additive. Suppose $I$ is (possibly) infinite. Then $\sum_{i \in I} A_i$ is a filtered colimit of all its finite "sub-coproducts". But for any finite subset $I' \subseteq I$, the canonical morphism $\sum_{i \in I'} A_i \cong \prod_{i \in I'} A_i \to \prod_{i \in I} A_i$ is a (split!) monomorphism, and filtered colimits preserve monomorphisms, so the canonical morphism $\sum_{i \in I} A_i \to \prod_{i \in I} A_i$ is indeed a monomorphism.
In fact, all we really needed was Grothendieck's axiom AB5 (in addition to AB3 and AB3*, of course), i.e. that filtered colimits in $\mathcal{A}$ are exact.
This statement isn't true in all abelian categories. The morphism $\bigoplus \mathbb{Z} \to \prod \mathbb{Z}$ from an infinite coproduct to an infinite product of copies of $\mathbb{Z}$ is monic but not epic in $\text{Ab}$, so the corresponding morphism in $\text{Ab}^{op}$ is epic but not monic. Freyd-Mitchell doesn't help here because the embedding it gives is only guaranteed to be exact; it isn't guaranteed to preserve infinite products or coproducts.
(Note that by Pontrjagin duality $\text{Ab}^{op}$ is the category of compact Hausdorff abelian groups, with $\mathbb{Z}$ being sent to $S^1$, so the claim is that the canonical morphism from an infinite coproduct to an infinite product of copies of $S^1$ is epic but not monic. This might seem strange, but the infinite coproduct in compact Hausdorff abelian groups is not the infinite direct sum: it's the Bohr compactification of the infinite direct sum. Also note that $\text{Ab}$ is locally finitely presentable and $\text{Ab}^{op}$ isn't, so maybe that's a hint as to what the correct conditions to impose are?)
